Question

If$\sum _{k=1}^{\infty }{a}_k$and $\sum _{k=1}^{\infty }{b}_k$are convergent series and c is any real number ,then

a)$\sum _{k=1}^{\infty }c{a}_k=c\sum _{k=1}^{\infty }{a}_k$

b)$\sum _{k=1}^{\infty }(a_k+b_k)=\sum _{k=1}^{\infty }{a}_k+\sum _{k=1}^{\infty }{b}_k$

Short answer

This means that adding or removing any finite number of terms from the start of the series does not change its convergent series

Step-by-step solution

the  given data

A) since we already know that both limits and sums commute with sums and constant multiples we will prove part b and leave the proof of part a )

B) given information ∑k=1∞(ak+bk)=∑k=1∞ak+∑k=1∞bk

The result follows from looking at the limits of partial sums

$$\begin{gathered} \sum _{k=1}^{\infty }(a_k+b_K)=\underset{n\to \infty }{\lim }\sum _{k=1}^{\infty }(a_k+b_k) \\ =\underset{n\to \infty }{\lim }(\sum _{k=1}^{\infty }{a}_k+\sum _{k=1}^{\infty }{b}_k) \\ =\underset{n\to \infty }{\lim }\sum _{k=1}^{\infty }{a}_k+\underset{n\to \infty }{\lim }\sum _{k=1}^{\infty }{b}_k \\ =\sum _{k=1}^{\infty }{a}_k+\sum _{k=1}^{\infty }{b}_k \end{gathered}$$

hence proved