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

Short 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-by-step solution

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 $\sum _{k=1}^9{a}_k$.