Question

Does the series converge or diverge? `sum_(n=0) ^oo (5^(n-1))/6^(n+1)`

`sum_(n=0) ^oo (5^(n-1))/6^(n+1)`

Answer 1

You can rewrite the series as

`sum_(n =0)^oo 5^(n - 1)/6^(n + 1) = (5^n5^-1)/(6^n6^1) = 1/30(5/6)^n`

Note that this is a geometric series with first term `1/30` and common ratio `5/6`. Since the common ratio is less than `1`, this series converges.

Hopefully this helps!