n Only in the Power

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,

= (4*2^n)/(3*3^n)=4/3 * (2/3)^n

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 nth root will conveniently rid the term of an nth 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

= 1/3 < 1

so the series converges.

Copyright © 1996 Department of Mathematics, Oregon State University

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