Step by step
The question
Question: decide if the series converges: the sum of ((3n + 1)/(n + 2))ⁿ for n from 1 to infinity.
Σn = 1∞(3n + 1n + 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 (3n + 1)/(n + 2).
n√an = 3n + 1n + 2
2 The limit
Divide top and bottom by n.
L = lim 3 + 1n1 + 2n
The parts with n at the bottom go to 0: L = 3.
= 3 + 01 + 0 = 3
3 Decide
L = 3 is bigger than 1: the series diverges.
= diverges
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: 5, 10, 20 terms add up to 96.8229, 20047.4283, huge.
It grows without end.
Right!
S5 ≈ 96.8229, S10 ≈ 20047.4283, S20 ≈ huge
Let's look back.
First: take the nth root.
Then: the limit.
Then: decide.
Last: check.
Answer
= diverges