91Ó°ÊÓ

We are given two series: $\underset{k=1}{\overset{∞}{∑}}{a}_k$and $\underset{k=M}{\overset{∞}{∑}}{a}_k$.

We need to show that either both the series converges or both diverge.

And also if$\underset{k=M}{\overset{∞}{∑}}{a}_k$converges to$L$, then$\underset{k=1}{\overset{∞}{∑}}{a}_k$converges to$a_1+a_2+a_3+....+a_{M-1}+L.$

Let us assume that the series $\underset{k=1}{\overset{∞}{∑}}{a}_k$is convergent. Then we have to show that the series $\underset{k=M}{\overset{∞}{∑}}{a}_k$is also convergent.

The series $\underset{k=1}{\overset{∞}{∑}}{a}_k$is convergent and converges to $A$. Therefore, the sequence $\left\{A_n\right\}$of the partial sum of the series $\underset{k=1}{\overset{∞}{∑}}{a}_k$is convergent and converges to $A$.

Let the sequence $\left\{B_n\right\}$be the sequence of the partial sum of the series role="math" localid="1652719100678" $\underset{k=M}{\overset{∞}{∑}}{a}_k$.

Therefore, by the definition of convergence of a sequence, for given $\mathrm{Î\mu }>0$, there exist a positive integer $N$such that $\left|A_n-A\right|<\mathrm{Î\mu }$for $kâ‰\yen N$......(1)

For $kâ‰\yen M$, the terms of the sequences $\left\{A_n\right\}$and $\left\{B_n\right\}$are the same.

Choose $P=max\left\{N,M\right\}$.

Therefore, role="math" localid="1652719052518" $\left|B_n-A\right|<\mathrm{Î\mu }$for $kâ‰\yen M$......(2)

Therefore, for given $\mathrm{Î\mu }>0$ there exist a positive integer P such that $\left|B_n-A\right|<\mathrm{Î\mu }$for $kâ‰\yen M$.

Hence, the sequence $\left\{B_n\right\}$of partial sum of the series $\underset{k=M}{\overset{∞}{∑}}{a}_k$is convergent and converges to $A$.

Thus, the series$\underset{k=M}{\overset{∞}{∑}}{a}_k$is convergent.

It is given that the series $\underset{k=M}{\overset{∞}{∑}}{a}_k$is convergent and converges to $L$.

We need to show that the series $\underset{k=1}{\overset{∞}{∑}}{a}_k$converges to $a_1+a_2+a_3+...+a_{M-1}+L$.

The series $\underset{k=1}{\overset{∞}{∑}}{a}_k$can be written as

$$\begin{gathered} \underset{k=1}{\overset{∞}{∑}}{a}_k=\underset{k=1}{\overset{M-1}{∑}}{a}_k+\underset{k=M}{\overset{∞}{∑}}{a}_k \\ \underset{k=1}{\overset{∞}{∑}}{a}_k=a_1+a_2+a_3+...+a_{M-1}+\underset{k=M}{\overset{∞}{∑}}{a}_k \\ \underset{k=1}{\overset{∞}{∑}}{a}_k=a_1+a_2+a_3+...+a_{M-1}+L \end{gathered}$$

So the series converges to $a_1+a_2+a_3+...+a_{M-1}+L$.