Chapter 9

Give examples to show that if \(\sum_{k} a_{k}\) and \(\sum_{k} b_{k}\) are convergent series of real numbers, then the series \(\sum_{k} a_{k} b_{k}\) may not be convergent. Also show that if \(\sum_{k} a_{k}=A\) and \(\sum_{k} b_{k}=B\), then \(\sum_{k} a_{k} b_{k}\) may be convergent, but its sum may not be equal to \(A B\).

Consider a series \(\sum_{k} a_{k}\) and for each \(n=0,1,2, \ldots\), let \(b_{n, k}:=a_{n+k} .\) Show that the series \(\sum_{k} a_{k}\) is convergent if and only if for some \(n=0,2, \ldots\), the series \(\sum_{k} b_{n, k}\) is convergent. In this case, prove that the series \(\sum_{k} b_{n, k}\) is convergent for every \(n=0,1,2, \ldots\), and \(\sum_{k} b_{n, k}=\sum_{k} a_{k}-A_{n}\), where \(A_{n}\) is the \(n\) th partial sum of the series \(\sum_{k} a_{k}\).

Show that the series \(\sum_{k=1}^{\infty} 2 /(k+1)(2 k+1)\) is convergent and its sum is less than or equal to 1. (Hint: Compare the given series with the series \(\left.\sum_{k=1}^{\infty} 1 / k(k+1) .\right)\)

Let \(a \in \mathbb{R}\) with \(a>1\). Show that the series \(\sum_{k=1}^{\infty}\left(1 / a^{k !}\right)\) is convergent.

Let \(a_{k} \in \mathbb{R}\) with \(a_{k} \leq 0\) for all \(k \in \mathbb{N}\). Show that \(\sum_{k=1}^{\infty} a_{k}\) is convergent if and only if the sequence \(\left(A_{n}\right)\) of its partial sums is bounded below, and in this case \(\sum_{k=1}^{\infty} a_{k}=\inf \left\\{A_{n}: n \in \mathbb{N}\right\\} .\) If \(\left(A_{n}\right)\) is not bounded below, then show that \(\sum_{k-1}^{\infty} a_{k}\) diverges to \(-\infty\).

Let \(\left(a_{k}\right)\) be a monotonically decreasing sequence of nonnegative real numbers. Show that the series \(\sum_{k=1}^{\infty} a_{k}\) is convergent if and only if the series \(\sum_{k=0}^{\infty} 2^{k} a_{2^{k}}\) is convergent. (Hint: Proposition 9.4.) Deduce the convergence and divergence of the series \(\sum_{k=1}^{\infty} 1 / k^{p}\) and \(\sum_{k=2}^{\infty} 1 / k(\ln k)^{p}\), where \(p \in \mathbb{R}\). (Compare Example 9.40.)

Suppose \(\left(a_{k}\right)\) is a monotonically decreasing sequence of nonnegative real numbers. If the series \(\sum_{k} a_{k}\) is convergent, then show that \(k a_{k} \rightarrow 0\) as \(k \rightarrow \infty\). (Hint: Exercise 7.) Also, show that the converse of this result does not hold.

A sequence \(\left(a_{k}\right)\) is said to be of bounded variation if \(\sum_{k=1}^{\infty}\left|a_{k}-a_{k+1}\right|\) is convergent. Prove the following: (i) A sequence of bounded variation is convergent. (ii) Let \(\left(a_{k}\right)\) and \(\left(b_{k}\right)\) be of bounded variation and let \(r \in \mathbb{R}\). Then \(\left(a_{k}+b_{k}\right)\), \(\left(r a_{k}\right)\), and \(\left(a_{k} b_{k}\right)\) are of bounded variation. If \(a_{k} \neq 0\) for all \(k \in \mathbb{N}\), is \(\left(1 / a_{k}\right)\) of bounded variation? (iii) Every bounded monotonically increasing sequence is of bounded variation. Further, if \(\left(b_{k}\right)\) and \(\left(c_{k}\right)\) are bounded monotonically increasing sequences and we define \(a_{k}:=b_{k}-c_{k}\) for \(k \in \mathbb{N}\), then the sequence \(\left(a_{k}\right)\) is of bounded variation. (iv) If \(\left(a_{k}\right)\) is of bounded variation, then there are bounded monotonically increasing sequences \(\left(b_{k}\right)\) and \(\left(c_{k}\right)\) such that \(a_{k}=b_{k}-c_{k}\) for \(k \in \mathbb{N}\). (Hint: Let \(a_{0}:=0\) and \(v_{k}:=\left|a_{1}\right|+\left|a_{1}-a_{2}\right|+\cdots+\left|a_{k-1}-a_{k}\right|\) for \(k \in \mathbb{N}\). Define \(b_{k}:=\left(v_{k}+a_{k}\right) / 2\) and \(c_{k}:=\left(v_{k}-a_{k}\right) / 2\) for \(k \in \mathbb{N} .\) )

(Ratio Comparison Test) Let \(\left(a_{k}\right)\) and \(\left(b_{k}\right)\) be sequences and suppose \(b_{k}>0\) for all \(k\). Prove the following: (i) If \(\left|a_{k+1}\right| b_{k} \leq\left|a_{k}\right| b_{k+1}\) for all large \(k\) and \(\sum_{k=1}^{\infty} b_{k}\) is convergent, then \(\sum_{k=1}^{\infty} a_{k}\) is absolutely convergent. (ii) If \(\left|a_{k+1}\right| b_{k} \geq\left|a_{k}\right| b_{k+1}\) for all large \(k\) and \(\sum_{k=1}^{\infty} b_{k}\) is divergent, then \(\sum_{k=1}^{\infty} a_{k}\) is not absolutely convergent.

Let \(a, b \in \mathbb{R}\) be such that \(01\).

For \(k \in \mathbb{N}\), let \(a_{2 k-1}:=4^{k-1} / 9^{k-1}\) and \(a_{2 k}:=4^{k-1} / 9^{k} .\) Show that \(\left|a_{2 k} / a_{2 k-1}\right|=\frac{1}{9}\) and \(\left|a_{2 k+1} / a_{2 k}\right|=4\) for all \(k \in \mathbb{N}\) and so the Ratio Test for the convergence of \(\sum_{k=1}^{\infty} a_{k}\) is inconclusive. Prove that \(\left|a_{k}\right|^{1 / k} \rightarrow \frac{2}{3}\) as \(k \rightarrow \infty\) and use the Root Test to conclude that \(\sum_{k=1}^{\infty} a_{k}\) is convergent.

Let \(\left(a_{k}\right)\) be a sequence of real numbers. If there is \(p>1\) such that $$ \left|a_{k+1}\right| \leq\left(1-\frac{p}{k}\right)\left|a_{k}\right| \quad \text { for all large } k, $$ then show that \(\sum_{k=1}^{\infty} a_{k}\) is absolutely convergent. On the other hand, if \(\left|a_{k+1}\right| \geq\left(1-\frac{1}{k}\right)\left|a_{k}\right| \quad\) for all large \(k\), then show that \(\sum_{k=1}^{\infty} a_{k}\) is divergent. (Hint: If \(p>1\) and \(x \in[0,1]\), then \(1-p x \leq(1-x)^{p}\). Use Exercise 10.)

(i) If \(a_{1}:=1\) and \(a_{k+1}:=(k-1) a_{k} /(k+1)\) for \(k \geq 2\), then show that \(\sum_{k=1}^{\infty} a_{k}\) is convergent. (ii) If \(a_{1}:=1\) and \(a_{k+1}:=(2 k-1) a_{k} / 2 k\) for \(k \in \mathbb{N}\), then show that \(\sum_{k=1}^{\infty} a_{k}\) diverges to \(\infty\)

Let \(\alpha, \beta, \gamma\) be positive real numbers. If \(a_{0}:=1\) and $$ a_{k}:=\frac{\alpha(\alpha+1) \cdots(\alpha+k-1) \beta(\beta+1) \cdots(\beta+k-1)}{\gamma(\gamma+1) \cdots(\gamma+k-1) k !} \text { for } k \in \mathbb{N} \text { , } $$ then show that \(\sum_{k=0}^{\infty} a_{k}\) is convergent if and only if \(\gamma>\alpha+\beta\). (Hint: Exercise 13.)

Suppose the partial sums of a series \(\sum_{k=1}^{\infty} b_{k}\) are bounded. If \(p>0\) and \(x \in(0,1)\), then show that the series $$ \sum_{k=1}^{\infty} \frac{b_{k}}{k^{p}}, \quad \sum_{k=1}^{\infty} \frac{b_{k}}{(\ln k)^{p}} \quad \text { and } \quad \sum_{k=1}^{\infty} b_{k} x^{k} $$ are convergent. (Hint: Proposition \(9.20 .\) )

If \(\left(a_{k}\right)\) is a bounded monotonic sequence and \(\sum_{k=1}^{\infty} b_{k}\) is a convergent series, then show that the series \(\sum_{k=1}^{\infty} a_{k} b_{k}\) is convergent.

Let \(\sum_{k=1}^{\infty} b_{k}\) be a convergent series. Show that the series $$ \sum_{k=1}^{\infty} k^{1 / k} b_{k} \quad \text { and } \quad \sum_{k=1}^{\infty}\left(1+\frac{1}{k}\right)^{k} b_{k} $$ are also convergent. (Hint: Exercise 17 and Exercises 7,8 of Chapter 2.)

Let \(p \in \mathbb{R}\) with \(p>1\). Show that $$ \frac{1}{(p-1)(\ln 2)^{p-1}} \leq \sum_{k=2}^{\infty} \frac{1}{k(\ln k)^{p}} \leq \frac{p-1+2 \ln 2}{2(p-1)(\ln 2)^{p}} $$ (Hint: Proposition 9.39 and Example 9.40 (ii).)

Find the radius of convergence of the power series \(\sum_{k=0}^{\infty} c_{k} x^{k}\) whose coefficients are defined by \(c_{2 k-1}:=3^{-k}\) and \(c_{2 k}:=2^{k} 5^{-k}\) for \(k \in \mathbb{N}\).

Find the radius of convergence of the power series \(\sum_{k=0}^{\infty} c_{k} x^{k}\) if for \(k \in \mathbb{N}\), the coefficient \(c_{k}\) is given as follows: (i) \(k !\), (ii) \(k^{2}\), (iii) \(\frac{k}{k^{2}+1}\), (iv) \(k e^{-k}\), (v) \(c^{k^{2}}\), where \(c \in \mathbb{R}\), (vi) \(\frac{k^{k}}{k !}\) (vii) \(\frac{2^{k}}{k^{2}}\) (viii) \(\left(\begin{array}{c}k+m \\ k\end{array}\right)\), where \(m \in \mathbb{N}\).

Let \(f(x):=\cos x\) for \(x \in \mathbb{R}\). Show that the Taylor series of \(f\) is convergent for \(x \in \mathbb{R}\). Deduce that $$ \cos x=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\cdots=\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{2 k}}{(2 k) !} \quad \text { for } x \in \mathbb{R} $$

Let \(a \in \mathbb{R}\) and \(I\) be an open interval containing \(a\). Show that if \(\sum_{k=0}^{\infty} a_{k}(x-a)^{k}\) is the Taylor series of some \(f: I \rightarrow \mathbb{R}\) around \(a\), then there are infinitely many functions \(g: I \rightarrow \mathbb{R}\) that have the same Taylor series around \(a\).

Let \(a \in \mathbb{R}\) and \(f:[a, \infty) \rightarrow \mathbb{R}\) be such that \(f(t) \leq 0\) for all \(t \geq a\) and \(f\) is integrable on \([a, x]\) for each \(x \geq a\). Show that \(\int_{a}^{\infty} f(t)\) is convergent if and only if its partial integral \(F:[a, \infty) \rightarrow \mathbb{R}\) defined by \(F(x):=\int_{a}^{x} f(t) d t\) is bounded below, and in this case \(\int_{a}^{\infty} f(t)=\inf \\{F(t): t \in[a, \infty)\\} .\) If \(F\) is not bounded below, then show that \(\int_{0}^{\infty} f(t)\) diverges to \(-\infty\).

Let \(f, g:[2, \infty) \rightarrow \mathbb{R}\) be defined by $$ f(t):=\left\\{\begin{array}{ll} 1 & \text { if } k \leq t

Let \(a \in \mathbb{R}\) and \(f:[a, \infty) \rightarrow \mathbb{R}\) be such that \(f\) is integrable on \([a, x]\) for all \(x \geq a\). Prove the following: (i) If \(\int_{a}^{\infty} f(t) d t\) is convergent and \(f(x) \rightarrow \ell\) as \(x \rightarrow \infty\), then \(\ell=0\). (ii) If \(f\) is differentiable and \(\int_{a}^{\infty} f^{\prime}(t) d t\) is convergent, then there is \(\ell \in \mathbb{R}\) such that \(f(x) \rightarrow \ell\) as \(x \rightarrow \infty .\) (Hint: Use part (i) of Proposition 6.21.) (iii) If \(f\) is differentiable and both \(\int_{a}^{\infty} f(t) d t\) and \(\int_{a}^{\infty} f^{\prime}(t) d t\) are convergent, then \(f(x) \rightarrow 0\) as \(x \rightarrow \infty\).

Show that \(\int_{1}^{\infty}\left(\cos t / t^{p}\right) d t\) and \(\int_{1}^{\infty}\left(\sin t / t^{p}\right) d t\) are absolutely convergent if \(p>1\) and that they are conditionally convergent if \(0

Let \(f:[1, \infty) \rightarrow \mathbb{R}\) be such that \(f\) is integrable on \([1, x]\) for every \(x \geq 1\). Prove the following using Proposition 9.42: (i) If there are \(p>1\) and \(\ell \in \mathbb{R}\) such that \(t^{p} f(t) \rightarrow \ell\) as \(t \rightarrow \infty\), then \(\int_{1}^{\infty} f(t) d t\) is absolutely convergent. (ii) Suppose \(f(t)>0\) for all \(t \in[1, \infty)\). If there are \(p \leq 1\) and \(\ell \neq 0\) such that \(t^{p} f(t) \rightarrow \ell\) as \(t \rightarrow \infty\), then \(\int_{1}^{\infty} f(t) d t\) is divergent.

Let \(g:[1, \infty) \rightarrow \mathbb{R}\) be a continuous real-valued function such that the function \(G:[a, \infty) \rightarrow \mathbb{R}\) defined by \(G(x):=\int_{a}^{x} g(t) d t\) is bounded. If \(p \in \mathbb{R}\) with \(p>0\) and \(x \in(0,1)\), then show that the improper integrals $$ \int_{1}^{\infty} \frac{g(t)}{t^{p}} d t, \quad \int_{1}^{\infty} \frac{g(t)}{(\ln t)^{p}} d t, \quad \text { and } \quad \int_{1}^{\infty} x^{t} g(t) d t $$ are convergent. (Hint: Proposition \(9.51 .\) )

Let \(a \in \mathbb{R}\) and \(f, g:[a, \infty) \rightarrow \mathbb{R}\) be such that \(f\) is bounded, monotonic, and differentiable, \(f^{\prime}\) is integrable on \([a, x]\) for every \(x \geq a, g\) is continuous, and \(\int_{a}^{\infty} g(t) d t\) is convergent. Show that \(\int_{a}^{\infty} f(t) g(t) d t\) is convergent. (Hint: Use Integration by Parts.) [Note: Compare with Proposition \(9.51\).

Let \(\int_{1}^{\infty} g(t) d t\) be a convergent improper integral. Show that the improper integrals \(\int_{1}^{\infty} t^{1 / t} g(t) d t\) and \(\int_{1}^{\infty}\left(1+\frac{1}{t}\right)^{t} g(t) d t\) are also convergent. (Hint: Exercise 34 , and Revision Exercise 15 given at the end of Chapter 7 .)

Let \(a \in \mathbb{R}\) and let \(f, g\) : \([a, \infty) \rightarrow \mathbb{R}\) be any functions. (i) If \(f\) is differentiable, \(\int_{a}^{\infty}\left|f^{\prime}(t)\right| d t\) is convergent, \(f(x) \rightarrow 0\) as \(x \rightarrow \infty\), \(g\) is continuous, and the function \(G:[a, \infty) \rightarrow \mathbb{R}\) defined by \(G(x):=\) \(\int_{a}^{x} g(t) d t\) is bounded, then show that \(\int_{a}^{\infty} f(t) g(t) d t\) is convergent. (ii) If \(f\) is differentiable, \(\int_{a}^{\infty}\left|f^{\prime}(t)\right| d t\) is convergent, \(g\) is continuous, and \(\int_{a}^{\infty} g(t) d t\) is convergent, then show that \(\int_{a}^{\infty} f(t) g(t) d t\) is convergent. (Hint: Use Integration by Parts.)

Show that the improper integrals \(\int_{1}^{\infty} \sin t^{2} d t\) and \(\int_{1}^{\infty} \cos t^{2} d t\) are convergent. (Hint: Substitute \(s=t^{2}\) and use Corollary 9.52.)

Let \(p \in \mathbb{R}\) with \(p>0\). Show that the improper integrals $$ \int_{2}^{\infty} \frac{\sin t}{(\ln t)^{p}} d t \text { and } \int_{2}^{\infty} \frac{\cos t}{(\ln t)^{p}} d t $$ are conditionally convergent. (Hint: Corollary \(9.52\) and Exercise 31 .)

Let \(f:[0, \infty) \rightarrow \mathbb{R}\) be such that \(\int_{0}^{\infty} f(t) d t\) is absolutely convergent. Show that the improper integrals $$ \begin{aligned} \mathcal{L}(f)(u) &:=\int_{0}^{\infty} f(t) e^{-u t} d t, \quad \text { where } u \in \mathbb{R} \text { and } u \geq 0, \\ \mathcal{F}_{s}(f)(u) &:=\frac{2}{\pi} \int_{0}^{\infty} f(t) \sin u t d t, \quad \text { where } u \in \mathbb{R} \\ \mathcal{F}_{c}(f)(u) &:=\frac{2}{\pi} \int_{0}^{\infty} f(t) \cos u t d t, \quad \text { where } u \in \mathbb{R} \end{aligned} $$ are absolutely convergent. [Note: \(\mathcal{L}(f), \mathcal{F}_{s}(f)\), and \(\mathcal{F}_{c}(f)\) are called the Laplace Transform, the Fourier Sine Transform, and the Fourier Cosine Transform of \(f .]\)

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(t):=(1+t) /\left(1+t^{2}\right)\) for \(t \in \mathbb{R}\). Show that \(\int_{-\infty}^{\infty} f(t) d t\) is divergent, but the Cauchy principal value of the integral of \(f\) on \(\mathbb{R}\) exists and is equal to \(\pi\).

Let \(a, b \in \mathbb{R}\) be such that \(a

Let \(f:[1, \infty) \rightarrow \mathbb{R}\) be defined as follows. If \(t \in[1, \infty)\) and \(k \leq t

Show that the improper integral \(\int_{0}^{\infty} e^{t^{2}} d t\) is divergent, but the improper integral \(\int_{0}^{\infty} e^{-t^{2}} d t\) is convergent. (Hint: Comparison with \(\int_{0}^{\infty} e^{t} d t\) and \(\left.\int_{0}^{\infty} e^{-t} d t\right)\)

Let \(p(t)\) and \(q(t)\) be polynomials of degrees \(m\) and \(n\) respectively. Suppose \(q(t) \neq 0\) for all \(t \geq a\) and let \(f:[a, \infty) \rightarrow \mathbb{R}\) be defined by \(f(t):=p(t) / q(t)\) Show that \(\int_{a}^{\infty} f(t) d t\) is absolutely convergent if \(n \geq m+2\) and \(\int_{a}^{\infty} f(t) d t\) is divergent if \(n

Let \(f:(a, b] \rightarrow \mathbb{R}\) be a nonnegative function that is integrable on \([x, b]\) for every \(x \in(a, b]\). Prove the following: (i) If there is \(p \in(0,1)\) such that \((t-a)^{p} f(t) \rightarrow \ell\) for some \(\ell \in \mathbb{R}\), then \(\int_{a^{+}}^{b} f(t) d t\) is convergent. (ii) If there is \(p \geq 1\) such that \((t-a)^{p} f(t) \rightarrow \ell\) for some \(\ell \neq 0\), then \(\int_{a^{+}}^{b} f(t) d t\) is divergent. (Hint: Corollary \(9.44\) with \(g(t):=1 /(t-a)^{p}\) for \(\left.t \in(a, b] .\right)\)

Show that \(\int_{1^{+}}^{2}(\sqrt{t} / \ln t) d t\) is divergent. (Hint: Exercise 45 .)

Let \(p, q \in \mathbb{R}\) with \(p>0\) and \(q>0 .\) Show that $$ \beta(p, q)=2 \int_{0^{+}}^{(\pi / 2)^{-}}(\sin u)^{2 p-1}(\cos u)^{2 q-1} d u $$ and in particular, \(\beta\left(\frac{1}{2}, \frac{1}{2}\right)=\pi .\) (Hint: Substitute \(\left.t:=\sin ^{2} u .\right)\)

Let \(p, q>0\). Show that $$ \beta(p, q)=\int_{0^{+}}^{\infty} \frac{u^{p-1}}{(1+u)^{p+q}} d u=\int_{0^{+}}^{1} \frac{v^{p-1}+v^{q-1}}{(1+v)^{p+q}} d v . $$ (Hint: Substitute \(t:=u /(1+u)\) and then \(v:=1 / u\).)

Show that \(\Gamma(s)=2 \int_{0}^{\infty} e^{-u^{2}} u^{2 s-1} d u\) for all \(s>0\) and in particular, that \(\Gamma\left(\frac{1}{2}\right)=2 \int_{0}^{\infty} e^{-u^{2}} d u .\) (Hint: Substitute \(\left.t:=u^{2} .\right)\)

Test the following for absolute/conditional convergence: (i) \(\int_{1}^{\infty} \frac{1}{\sqrt{1+t^{3}}} d t\), (ii) \(\int_{1}^{\infty} \frac{t^{q}}{1+t^{q}} d t\), where \(q \in \mathbb{R}\), (iii) \(\int_{2}^{\infty} \frac{1}{\ln t} d t\), (iv) \(\int_{0^{+}}^{1} \sin \left(\frac{1}{t}\right) d t\), (v) \(\int_{0^{+}}^{1} e^{1 / t} t^{q} d t\), where \(q \in \mathbb{R}\), (vi) \(\int_{0^{+}}^{1-} \frac{1}{t \ln t} d t\).

Suppose one of the series \(\sum_{k=0}^{\infty} a_{k}\) and \(\sum_{k=0}^{\infty} b_{k}\) is absolutely convergent and the other is convergent. Let \(A\) and \(B\) denote their respective sums. For each \(k=0,1, \ldots\), let \(c_{k}:=\sum_{j=0}^{k} a_{j} b_{k-j} .\) Show that the series \(\sum_{k=0}^{\infty} c_{k}\) is convergent and its sum is equal to \(A B\). Give an example to show that the result may not hold if both the series \(\sum_{k=0}^{\infty} a_{k}\) and \(\sum_{k=0}^{\infty} b_{k}\) are conditionally convergent.

Let \(m_{0}:=0\) and \(m_{1}

Let \(k \longmapsto j(k)\) be a bijection from \(\mathbb{N}\) to \(\mathbb{N}\). Given a series \(\sum_{k=1}^{\infty} a_{k}\), consider the series \(\sum_{k=1}^{\infty} b_{k}\), where \(b_{k}:=a_{j(k)}\) Then the series \(\sum_{k=1}^{\infty} b_{k}\) is called a rearrangement of the series \(\sum_{k=1}^{\infty} a_{k}\) Show that a series \(\sum_{k=1}^{\infty} a_{k}\) is absolutely convergent if and only if every rearrangement of it is convergent. In this case, the sum of a rearrangement is unchanged.

Use the triangle inequality and the Cauchy Criterion (Propositions \(1.8\) and \(2.19\) ) to conclude that if a series of real numbers is absolutely convergent, then it is convergent. Conversely, assuming that every absolutely convergent series of real numbers is convergent, deduce the Cauchy Criterion. (Hint: Given a Cauchy sequence \(\left(A_{n}\right)\) of real numbers, inductively construct a subsequence \(\left(A_{n_{k}}\right)\) such that \(\left|A_{n_{k+1}}-A_{n_{k}}\right| \leq 1 / k^{2}\) for all \(k \in \mathbb{N}\) and consider \(\left.a_{k}:=A_{n_{k+1}}-A_{n_{k}} .\right)\)

For \(k \in \mathbb{N}\), let \(a_{k} \in \mathbb{R}\) with \(a_{k}>0 .\) Show that $$ \liminf _{k \rightarrow \infty} \frac{a_{k+1}}{a_{k}} \leq \liminf _{k \rightarrow \infty} a_{k}^{1 / k} \quad \text { and } \quad \limsup _{k \rightarrow \infty} a_{k}^{1 / k} \leq \limsup _{k \rightarrow \infty} \frac{a_{k+1}}{a_{k}} $$