Q. 1 - Calculus

Question

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.

Part (a): If the sequence $\left\{a_n\right\}$ converges, then the series $\sum _{n=1}^{\infty } a_n$ converges.

Part (b): If a series $\sum _{k=1}^{\infty } a_k$ diverges and $\left\{S_n\right\}$ is its sequence of partial sums, then $\underset{n\to \infty }{\lim } S_n=\infty$.

Part (c): If two series $\sum _{k=1}^{\infty } a_k,\sum _{k=1}^{\infty } b_k$ both diverge, then the series $\sum _{k=1}^{\infty } \left(a_k+b_k\right)$ diverges.

Part (d): If $\sum _{k=1}^{\infty } a_k ,\sum _{k=1}^{\infty } b_k$ are two convergent series, then for any real numbers c and d, $\sum _{k=1}^{\infty } \left(c{a}_k+d{b}_k\right)=c\sum _{k=1}^{\infty } a_k+d\sum _{k=1}^{\infty } b_k$.

Part (e): If the series $\sum _{k=0}^{\infty } a_{n+k}$ converges to $5$, then the series $\sum _{k=100}^{\infty } a_k$ converges to a value $L<5$.

Part (f): If the series $\sum _{k=N}^{\infty } a_k$ converges, then $\sum _{k=N}^{\infty } a_k=\sum _{k=0}^{\infty } a_{n+k}$.

Part (g): If a geometric series $\sum _{k=0}^{\infty } c{r}^k$ converges, then $\underset{k\to \infty }{\lim } c{r}^k=0$.

Part (h): The series $\sum _{k=1}^{\infty } a_k$ where $$\begin{gathered} a_1=4, \\ {a}_{k+1}=\frac{a_k}{2} \end{gathered}$$ for $k>1$, converges to $7$.

Step-by-Step Solution

Answer

Part (a): The given statement is false.

Part (b): The given statement is false.

Part (c): The given statement is false.

Part (d): The given statement is true.

Part (e): The given statement is false.

Part (f): The given statement is true.

Part (g): The given statement is true.

Part (h): The given statement is false.

Part (a) Step 1. Determine if the statement is true or false.

The sequence $\left\{a_n\right\}=\left\{\frac{1}{n}\right\}$ is a convergent sequence and converges to $0$.

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

Therefore, if the sequence $\left\{a_n\right\}$ converges, then $\sum _{k=1}^{\infty } a_n$ converges is not true.

Hence, the given statement is false.

Part (b) Step 1. Determine if the statement is true or false.

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

Consider the sequence $\left\{S_n\right\}=\left\{\frac{1}{n}\right\}$.

The sequence $\left\{a_n\right\}=\left\{\frac{1}{n}\right\}$ is a convergent sequence and converges to $0$.

Therefore, if the series $\sum _{k=1}^{\infty } a_k$ diverges, then $\left\{S_n\right\}$ is its sequence of partial sums, then $\underset{n\to \infty }{\lim }{S}_n=\infty$ is not true.

Hence, the given statement is false.

Part (c) Step 1. Determine if the statement is true or false.

Consider the geometric series,$\sum _{k=0}^{\infty } a_k=\sum _{k=0}^{\infty }1$.

The series $\sum _{k=0}^{\infty } 1$ is a geometric series with common ratio $r=1$, which is equal to $1$.

The geometric series with ratio equal to $1$ is divergent.

Therefore, $\sum _{k=0}^{\infty } a_k=\sum _{k=0}^{\infty }1$ is divergent.

Again, consider the geometric series, $\sum _{k=0}^{\infty } b_k=\sum _{k=0}^{\infty }\left(-1\right)$.

The series $\sum _{k=0}^{\infty } b_k=\sum _{k=0}^{\infty }\left(-1\right)$ is a geometric series with common ratio $r=1$, which is equal to $1$.

The geometric series with ratio equal to $1$ is divergent.

Therefore, $\sum _{k=0}^{\infty } b_k=\sum _{k=0}^{\infty }\left(-1\right)$ is divergent.

Part (c) Step 2. Consider the series &#8721; k = 0 &#8734; &#160; a k + b k .

Consider the series,$\sum _{k=0}^{\infty } \left(a_k+b_k\right)$.

$$\begin{gathered} \sum _{k=0}^{\infty } \left(a_k+b_k\right)=\sum _{k=0}^{\infty } \left(1+\left(-1\right)\right) \\ =\sum _{k=0}^{\infty } 0 \\ =0 \end{gathered}$$

The partial sum of series $\sum _{k=0}^{\infty } 0$ is a constant and hence, it is convergent.

Therefore, $\sum _{k=0}^{\infty } \left(a_k+b_k\right)=\sum _{k=0}^{\infty } 0$ is convergent.

Hence, the given statement is false.

Part (d) Step 1. Determine if the statement is true or false.

The series $\sum _{k=1}^{\infty } a_k ,\sum _{k=1}^{\infty } b_k$ are convergent.

Consider the series $\sum _{k=1}^{\infty } \left(c{a}_k+d{b}_k\right)$. Then,

$$\begin{gathered} \sum _{k=1}^{\infty } \left(c{a}_k+d{b}_k\right)=\left(c{a}_1+d{b}_2\right)+\left(c{a}_2+d{b}_2\right)+\left(c{a}_3+d{b}_3\right)+... \\ =\left(c{a}_1+c{a}_2+c{a}_3+...\right)+\left(d{b}_1+d{b}_2+d{b}_3+...\right) \\ =c\left(a_1+a_2+a_3+...\right)+d\left(b_1+b_2+b_3+...\right) \\ =\underset{k=1}{\overset{\infty }{c\sum }} a_k+d\sum _{k=1}^{\infty } b_k \end{gathered}$$

Hence, the given statement is true.

Part (e) Step 1. Determine if the statement is true or false.

As it is known the adding and deleting the terms from the series does not affect the convergence of the series.

Thus, if the series $\sum _{k=0}^{\infty } a_{n+k}$ converges to $5$, then the series $\sum _{k=0}^{\infty } a_{n+k}$ converges to a value $L<5$ is not true.

Hence, the given statement is false.

Part (f) Step 1. Determine if the statement is true or false.

As it is known the adding and deleting the terms from the series does not affect the convergence of the series.

Thus, if the series $\sum _{k=N}^{\infty } a_k$ converges, then $\sum _{k=N}^{\infty } a_k=\sum _{k=0}^{\infty } a_{n+k}$ is true.

Hence, the given statement is true.

Part (g) Step 1. Determine if the statement is true or false.

If a series is convergent, then the limit of the nth term is zero.

Thus, if a geometric series $\sum _{k=0}^{\infty } c{r}^k$ converges, then $\underset{k\to \infty }{\lim }c{r}^k=0$ is true.

Hence, the given statement is true.

Part (h) Step 1. Determine if the statement is true or false.

The terms of the series $\sum _{k=0}^{\infty } a_k$ are given below,

$\sum _{k=0}^{\infty } a_k\mathit{=}\mathit{4}\mathit{+}\frac{\mathit{4}}{\mathit{2}}\mathit{+}\frac{\mathit{4}}{{\mathit{2}}^{\mathit{2}}}\mathit{+}\mathit{.}\mathit{.}\mathit{.}$

The sum of the series is given below,

$$\begin{gathered} S_{\infty }=\frac{4}{1-{\displaystyle \frac{1}{2}}} \\ =8 \end{gathered}$$

Thus, if the series $\sum _{k=1}^{\infty } a_k$ where $$\begin{gathered} a_1=4, \\ {a}_{k+1}=\frac{a_k}{2} \end{gathered}$$ for $k>1$, converges to $7$ is not true.

Hence, the given statement is false.