Question

Does the series pi+ 3pi/4 +9pi/16 + 27pi/64... converges or diverges?

Answer 1

\[\begin{align*}\pi+\frac{3\pi}{4}+\frac{9\pi}{16}+\frac{27\pi}{64}+\cdots&=\pi\left(1+\frac{3}{4}+\frac{9}{16}+\frac{27}{64}+\cdots\right)\\
&=\pi\left(\left(\frac{3}{4}\right)^0+\left(\frac{3}{4}\right)^1+\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3+\cdots\right)\\
&=\pi\sum_{n=0}^\infty \left(\frac{3}{4}\right)^n
\end{align*}\]
What do you know about geometric series?

Answer 2

Not much ...what does coverges and diverges mean
?

Answer 3

For a sum to converge, that means that the more terms you add, the final sum won't keep increasing or decreasing, but rather approach a number.

For example, consider the sum
\[\begin{align*}1&=1\\
1+\frac{1}{10}&=1.1\\
1+\frac{1}{10}+\frac{1}{100}&=1.11\end{align*}\]
If we add more terms to follow the above pattern, we say the sum converges because the sum of infinite of these terms approaches a number:
\[1+\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\cdots=1+0.1+0.001+0.0001+\cdots=1.111...=\frac{10}{9}\]
On the other hand, a sum diverges if you add more terms and the sum doesn't approach a finite value. As an example consider the sum,
\[1+2+3+\cdots\]
As you keep adding terms in the pattern, the sum approaches infinity, which is not finite.