Question

Find the sum of the series when it converges. (if geometric, find the first term, the ratio and the sum)

Answer 1

\[\sum_{\infty}^{n=0}2(\frac{ 2 }{ 3 })^{n}\]

Answer 2

Ok, so you do know the formula for the sum of an infinite geometric series, it's actually one easy one to remember once you know it.

Answer 3

What is the 1st term?

Answer 4

im not sure the formula.

Answer 5

well do you mean the lim n->inf r^n = 1 if r<1 and 0 if |r|<1

Answer 6

$$
\Large{
\sum_{n=0}^\infty 2\left ( \cfrac{2}{3}\right )^n\\
=2\times\left ( \cfrac{2}{3}\right )^0+2\times\left ( \cfrac{2}{3}\right )^1+\cdots\\
=2\times1+2\times\left ( \cfrac{2}{3}\right )+\cdots\\
=2+\cfrac{4}{3}+\cdots\\
}
$$
Make sense so far?

Answer 7

Do you see the 1st term?

Answer 8

The terms in parenthesis above is the ratio

Answer 9

The formula for the sum uses this ratio:
$$
\sum_{n=0}^\infty 2\left ( \cfrac{2}{3}\right )^n\\
=2\left ( \cfrac{1}{1-a}\right )\\
$$
Where \(a\) is the ratio you found above.