Step by step
The question
Question: decide if the series converges: the sum of (n/(3n + 2))ⁿ for n from 1 to infinity.
Σn = 1∞(n3n + 2)n =
1 Take the nth root
The root test: take the nth root of the term, and let n grow.
If the limit L is less than 1, the series converges.
Bigger than 1: it diverges.
root test: L = lim n√an
L < 1 converges, L > 1 diverges
The nth root undoes the power n: what is left is n/(3n + 2).
n√an = n3n + 2
2 The limit
Divide top and bottom by n.
L = lim 13 + 2n
The parts with n at the bottom go to 0: L = 1/3.
= 13 + 0 = 13
3 Decide
L = 1/3 is less than 1: the series converges.
= converges
Careful!
It is the other way: L less than 1 means the terms shrink fast, like a geometric series.
It converges.
Careful: not L < 1, so it diverges ✗
4 Check
Let's check with a calculator: 10, 20, 50 terms add up to 0.2927, 0.2927, 0.2927.
It settles.
Right!
S10 ≈ 0.2927, S20 ≈ 0.2927, S50 ≈ 0.2927
Let's look back.
First: take the nth root.
Then: the limit.
Then: decide.
Last: check.
Answer
= converges