Question

Explain how to change the index of the series$\sum _{k=1}^{\infty }{a}_k$to start with an initial value other than$1$.

Short answer

To change the index of the given series,

$\sum _{k=1}^{\infty }{a}_k=\sum _{k=1}^{\infty }{a}_{j-s+1}$

Step-by-step solution

Step 1. Given information.

Consider the given question,

The series is$\sum _{k=1}^{\infty }{a}_k$.

Step 2. Change the index.

We need to change the index.

For $k=1$, substitute $k=j-s+1$,

$$\begin{gathered} 1=j-s+1 \\ j=s \end{gathered}$$

Therefore, the given series changes to$\sum _{k=1}^{\infty }{a}_k=\sum _{k=1}^{\infty }{a}_{j-s+1}$.