Q 60. - Calculus

Question

Determine whether the series $\frac{99}{40}-\frac{99}{20}+\frac{99}{10}-\frac{99}{5}+\dots$ converges or diverges. Give the sum of the convergent series.

Step-by-Step Solution

Answer

The series $\frac{99}{40}-\frac{99}{20}+\frac{99}{10}-\frac{99}{5}+\dots$ diverges.

Step 1. Given information.

Given a series $\frac{99}{40}-\frac{99}{20}+\frac{99}{10}-\frac{99}{5}+\dots$.

Step 2. Find if the series converges or not.

The standard form of a geometric series is $\sum _{k=0}^{\infty }c{r}^k$.

The geometric series converges if and only if $\left|r\right|<1$.

In the series $\frac{99}{40}-\frac{99}{20}+\frac{99}{10}-\frac{99}{5}+\dots$ it can be seen that $c=\frac{99}{40}$.

Every term after that is $-2$ times the previous term.

It follows that $r=-2$.

Since $r=-2$, the series $\frac{99}{40}-\frac{99}{20}+\frac{99}{10}-\frac{99}{5}+\dots$ diverges.