If some terms are negative, look at the series of absolute values instead. The Absolute Convergence Test says that if the latter series converges, then so does the former.

Try to isolate the part of the term raised to the *n* power from
the part of the series not raised to the *n* power. Consider splitting
the series into two pieces based on this, if necessary.

So, the sum of 3*(1/2^*n*) is 3 times the sum of 1/2^*n*. The
sum of 1/2^*n* converges, so 3 times is also converges. Similarly,
the sum of 3+1/2^*n* equals the sum of 3 + the sum of 1/2^*n*.
Since the sum of 3 diverges, and the sum of 1/2^*n* converges, the
series diverges. You have to be careful here, though: if you get a sum
of two diverging series, occasionally they will cancel each other out and
the result will converge.

If the power is *n*+1 or such, then factor out terms until you just
have an *n* power. Then try to match up the term with the
Geometric
Series: use the part raised to the *n* power as *r*, and the
part
not raised to the *n* power as *a*. If it doesn't fit exactly,
see if you can use a Comparison
Test or Limit Comparison
Test.

For instance,

the sum of which converges to 4/3*(1/(1-2/3)) = 4.

It also may be worthwhile to try the
Root Test, since taking an *n*th
root will conveniently rid the term of an *n*th power. Also, you will
often get a lot of cancellation using the
Ratio Test.

As an example, look at the sum of (1/3)^(*n*+2). Using the Ratio
Test, we get

so the series converges.

Copyright © 1996 Department of Mathematics, Oregon State University

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