Q. 1 - Calculus

Question

True/False: 

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If $a_k\to 0$, then $\sum _{k=1}^{\infty }{a}_k$ converges.

(b) True or False: If $\sum _{k=1}^{\infty }{a}_k$ converges, then $a_k\to 0$.

(c) True or False: The improper integral ${\int }_1^{\infty }f\left(x\right)dx$ converges if and only if the series $\sum _{k=1}^{\infty }f\left(k\right)$converges.

(d) True or False: The harmonic series converges.

(e) True or False: If $p>1$, the series $\sum _{k=1}^{\infty }{k}^{-p}$ converges.

(f) True or False: If $f\left(x\right)\to 0$ as $x\to \infty$, then $\sum _{k=1}^{\infty }f\left(k\right)$  converges.

(g) True or False: If $\sum _{k=1}^{\infty }f\left(k\right)$ converges, then $f\left(x\right)\to 0$ as $x\to \infty$.

(h) True or False: If $\sum _{k=1}^{\infty }{a}_k=L$ and $\left\{S_n\right\}$ is the sequence of partial sums for the series, then the sequence of remainders $\{L-S_n\}$ converges to $0$.

Step-by-Step Solution

Answer

(a) False.

(b) True.

(c) False.

(d) False.

(e) True.

(f) False.

(g) False.

(h) True.

Part (a) Step 1. Given Information.

A statement.

Part (a) Step 2. Consider the given series.

Consider the given series.

$$\begin{gathered} \sum _{k=1}^{\infty }{a}_k=\sum _{k=1}^{\infty }\frac{1}{k} \\ =\underset{k\to \infty }{lim}\frac{1}{k} \\ =0 \end{gathered}$$

Part (a) Step 3. Using harmonic series and p-test series.

So by using the harmonic series and p-test series, the series is divergent.

So the given statement is false.

Part (b) Step 1. Consider the given series.

Consider the given series,

$$\begin{gathered} \sum _{k=1}^{\infty }{a}_k=\sum _{k=1}^{\infty }\frac{1}{k^2} \\ =\underset{k\to \infty }{lim}\frac{1}{k^2} \\ =0 \end{gathered}$$

So the series is convergent and $a_k\to 0$.

So the statement is true.

Part (c) Step 1. Integral test.

If a(x) is a function that is continuous, positive, and decreasing on $[1,\infty )$, and if $\left\{a_k\right\}$ is the sequence defined by $a_k=a\left(k\right)$ for every $k\in {\mathrm{\mathbb{Z}}}^{+}$, then,

$\sum _{k=1}^{\infty }{a}_k and {\int }_1^{\infty }a\left(x\right) dx$

either both converge or both diverge.

Part (c) Step 2. Determine if the statement is true.

From the statement, it is false.

Part (d) Step 1. Consider the series.

Consider the series,

$$\begin{gathered} \sum _{k=1}^{\infty }{a}_k=\sum _{k=1}^{\infty }\frac{1}{k} \\ =\underset{k\to \infty }{lim}\frac{1}{k} \\ =0 \end{gathered}$$

Part (d) Step 2. Determine if the statement is true.

So by using the harmonic series and p-test series, the series $\sum _{k=1}^{\infty }\frac{1}{k}$ is divergent.

So the statement is false.

Part (e) Step 1. Consider the series.

Consider the series,

$\sum _{k=1}^{\infty }{a}_k=\sum _{k=1}^{\infty }\frac{1}{k^p}$

It is convergent by p-series when $p>1$.

So the statement is true.

Part (f) Step 1. Consider the function.

Consider the function,

$$\begin{gathered} f\left(x\right)=\frac{1}{x} \\ \underset{x\to \infty }{lim}f\left(x\right)=\underset{x\to \infty }{lim}\frac{1}{x} \\ =0 \end{gathered}$$

So by using the harmonic series and p-test series, the series is divergent and is not convergent.

So the given statement is false.

Part (g) Step 1. Consider the function.

Consider the function,

$$\begin{gathered} f\left(x\right)=\frac{x^2}{x^2+1} \\ \underset{x\to \infty }{lim}f\left(x\right)=\underset{x\to \infty }{lim}\frac{x^2}{x^2+1} \\ =\frac{1}{1+\frac{1}{x^2}} \\ =1 \end{gathered}$$

Part (g) Step 2. Determine if the statement is true.

So the series is a convergent series.

So the given statement is false.

Part (h) Step 1. Consider the sequence of remainders.

Consider the sequence of remainders,

$\{L-S_n\}$

$$\begin{gathered} \underset{n\to \infty }{lim}\{L-S_n\}=\underset{n\to \infty }{lim}\left\{L\right\}-\underset{n\to \infty }{lim}\left\{S_n\right\} \\ =L-L \\ =0 \end{gathered}$$
So the sequence convergent to zero.

So the statement is true.