How to determine whether \sum_{n=1}^oo (\frac{2}{n}+\frac{3}{2})^n converges or diverges?

1 Answer
turksvids
Jan 1, 2018

sum_(n=1)^oo(2/n+3/2)^n diverges by nth term test for divergence or direct comparison to divergent geometric series.

Explanation:

nth term test for divergence:
Since lim_(n to oo) (2/n+3/2)^n = oo ne 0, sum_(n=1)^oo(2/n+3/2)^n diverges by nth term test for divergence.

Direct Comparison to divergent geometric:
For n>=1, (2/n+3/2) > 3/2.
So we know that sum_(n=1)^oo(2/n+3/2)^n > sum_(n=1)^oo(3/2)^n
We know that sum_(n=1)^oo(3/2)^n is a divergent geometric series with r=3/2>1, therefore sum_(n=1)^oo(2/n+3/2)^n diverges by direct comparison.

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