Infinite series 1/(2+3^n)

Question

How do I prove the infinite series 1 / (2+3ⁿ) starting from n = 4 to infinity converges? I must use the ratio test but when I do so I am not getting a number less than 1.

Answer 1

Using the Ratio Test,

lim_n→∞[a_n+1 / a_n] = lim_n→∞[(2+3ⁿ) / (2+3ⁿ⁺¹)]

Dividing the numerator and denominator by 3ⁿ, we get

lim_n→∞[(2/3ⁿ + 1) / (2/3ⁿ + 3)] = (0 + 1) / (0 + 3) = 1/3 < 1

So, by the Ratio Test, the series is convergent.

Answer 2

If one test doesn't work, try another one! Use the root test which takes the limit as follows:

limit, n->infinity of (ABS(a_n))^(1/n)

or {1/(2+3n)}(1/n) you can start at 4 as you indicated but this appears to have a limit of 1/3 which is less

than 1 which indicates that the series converges!