Question

Given that $a_0=-3,a_1=5,a_2=-4,a_3=2$and $\underset{k=2}{\overset{\infty }{\sum a_k}}=7$, find the value of$\underset{k=0}{\overset{\infty }{\sum a_k}}$.

Short answer

The value of$\underset{k=0}{\overset{\infty }{\sum a_k}}=9$.

Step-by-step solution

Step 1. Given information.

Given that$a_0=-3,a_1=5,a_2=-4,a_3=2$and$\underset{k=2}{\overset{\infty }{\sum a_k}}=7$.

Step 2. Express the series in terms of known values.

Value of $\underset{k=0}{\overset{\infty }{\sum a_k}}$can be obtained as follows.

$\begin{array}{rcl}\underset{k=0}{\overset{\infty }{\sum a_k}}& =& a_0+a_1+\underset{k=2}{\overset{\infty }{\sum a_k}}\\ & =& -3+5+7\\ & =& 9\end{array}$

Therefore, value of$\sum _{k=0}^{\infty }{a}_k$is 9.