Q 46. - 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 $\underset{k=4}{\overset{\infty }{\sum a_k}}$.

Step-by-Step Solution

Answer

The value of $\underset{k=4}{\overset{\infty }{\sum a_k}}$ is 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 $\underset{k=4}{\overset{\infty }{\sum a_k}}$ can be obtained as follows.

$\begin{array}{rcl}\underset{k=4}{\overset{\infty }{\sum a_k}}& =& \underset{k=2}{\overset{\infty }{\sum a_k}}-\left(a_2+a_3\right)\\ & =& 7-\left(-4+2\right)\\ & =& 7+2\\ & =& 9\end{array}$

Therefore, value of $\sum _{k=4}^{\infty }{a}_k$ is 9.