Q 43. - Calculus

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 $\underset{k=0}{\overset{\infty }{\sum a_k}}$.

Step-by-Step Solution

Answer

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

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.