Question

Find the sum of the infinite series 1/3+4/9+16/27+64/81+... if it exists



16.75

30.55

8.99

none

Answer 1

no this formula is applicable only if |r|<1
when r=1 or>1
it approaches infinity

Answer 2

\[r^n \rightarrow \infty if ~r \geq1\]

Answer 3

\[
\begin{aligned}
\dfrac13+\dfrac49+\dfrac{16}{27}+\dfrac{64}{81}+\cdots&=\dfrac13\left(
1+\dfrac43+\dfrac{16}9+\dfrac{64}{27}+\cdots\right)\\
&=\dfrac13\sum_{n=0}^\infty\left(\dfrac43\right)^n
\end{aligned}
\]

Answer 4

\[\dfrac13\sum_{i=0}^n\left(\dfrac43\right)^i\ge\dfrac13\sum_{i=0}^n1\]
And clearly \(\displaystyle \lim_{n\to\infty}\dfrac13\sum_{i=0}^n1 = \infty\)

Therefore our summation diverges.