Question

How do you find the sum of `1+1/5+1/7^2+1/7^3+1/7^9+...+1/7^n+...`?

Answer 1

`1+1/5+1/7^2+1/7^3+1/7^9+...+1/7^n+... = 1+1/5+1/49+1/343 +7/6-282475242/242121642 ~= 1.223323644`

Explanation:

The question is not very clear...in answering I assume that for `n>=9` the sum is on all values of `n`

We can write the sum as:

`1+1/5+1/49+1/343 + sum_(n=9)^oo 1/7^n`

Focus on the series: clearly we have:

`sum_(n=9)^oo 1/7^n = sum_(n=0)^oo 1/7^n - sum_(n=0)^8 1/7^n`

Thus at the second member we have the sum of a geometric series and the partial sum of a geometric series of ratio `1/7`:

`sum_(n=0)^oo 1/7^n = sum_(n=0)^oo (1/7)^n = 1/(1-1/7) = 7/6`

`sum_(n=0)^8 1/7^n = sum_(n=0)^8 (1/7)^n = (1-1/7)^9/(1-1/7) = 7/6(1-1/40353607) = 282475242/242121642`

So:

`1+1/5+1/7^2+1/7^3+1/7^9+...+1/7^n+... = 1+1/5+1/49+1/343 +7/6-282475242/242121642 ~= 1.223323644`