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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