Q. 7 - Calculus

Question

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

Step-by-Step Solution

Answer

To change the index of the given series,

$\sum _{k=1}^{\infty } a_k=\sum _{k=1}^{\infty } a_{j-s+1}$

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 $\sum _{k=1}^{\infty } a_k=\sum _{k=1}^{\infty } a_{j-s+1}$.