Q. 5 - Calculus

Question

What is the relationship between $\sum _{k=1}^{\infty } a_k ,\sum _{k=10}^{\infty } a_k$? If you knew that $\sum _{k=10}^{\infty } a_k$ converged to $42$, what other information would you need to know to find the sum of $\sum _{k=1}^{\infty } a_k$?

Step-by-Step Solution

Answer

The required relationship is $\sum _{k=1}^{\infty } a_k=\sum _{k=1}^9 a_k+\sum _{k=10}^{\infty } a_k$.

To find the sum of $\sum _{k=10}^{\infty } a_k$, first it is required to know$\sum _{k=1}^9 a_k$.

Step 1. Given information.

Consider the given question,

The series $\sum _{k=1}^{\infty } a_k ,\sum _{k=10}^{\infty } a_k$ and the series $\sum _{k=10}^{\infty } a_k$ is converging to $42$.

Step 2.

The series $\sum _{k=1}^{\infty } a_k$ is written as,

$\sum _{k=1}^{\infty } a_k=\sum _{k=1}^9 a_k+\sum _{k=10}^{\infty } a_k ...... \left(i\right)$

To find the sum of $\sum _{k=1}^{\infty } a_k$, the sum of the series $\sum _{k=1}^{\infty } a_k$ is needed because from equation (i),

The sum of $\sum _{k=10}^{\infty } a_k$ is known and only value needed is $\sum _{k=1}^9 a_k$.