Determine whether the series converges or diverges. If it converges find what it converges to.

$\sum_{n=3}^\infty\frac{5^n + 2^n}{9^n}$. I tried the test for divergence but it is infinity over infinity which is an indeterminate form.

  • $\begingroup$ Try limit comparison with $\sum (5/9)^n$. $\endgroup$ – vadim123 Jul 7 '15 at 3:05
  • $\begingroup$ Alternatively, multiply by $\frac{9^n}{9^n}$ and simplify to get a summation you know converges. $\endgroup$ – user217285 Jul 7 '15 at 3:06

It is justifiable to split the sum up as $\displaystyle \sum_{n=0}^{\infty}\left(\frac{5}{9}\right)^n + \sum_{n=0}^{\infty}\left(\frac{2}{9}\right)^n$ as both series are absolutely convergent.

From that point on, just use the formula for geometric series.

  • $\begingroup$ I will just point out that $\sum(a_n+b_n)=\sum a_n+\sum b_n$ works for any convergent series, not only absolutely convergent. $\endgroup$ – Martin Sleziak Dec 4 '15 at 12:58

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