As an example,
Then use the Comparison Test, comparing with the p-series above:
If there aren't any transcendental functions (like the natural logarithm, the tangent function, and so forth) in the term, do the following: find the largest power of n contained in the a[n] term. If the term is a fraction, find the largest power of n in both the numerator and the denominator, and subtract the largest power in the denominator from the largest power in the numerator to get the largest power of n in the entire term. If two n terms are multiplied, add the powers. If a group of terms is inside, say, a cube root, divide all powers inside the cube root by three.
The resulting highest power should be negative. If not, then find the limit of the terms: it probably won't be zero. If it is negative, then try the Comparison Test or the Limit Comparison Test with b[n] = 1/n^p, where p is the power from above. If that doesn't work, and the term looks like something you can integrate, try the Integral Test.
has a power of 3 in the numerator, and a power of 2+2/3 in the denominator, so the whole fraction should compare favorably with n^(1/3). Use the Limit Comparison Test, and divide the above term by n^(1/3) to get
Then, in order to find the limit as n goes to infinity, divide both top and bottom by n^3 to get
and then the limit is found to be 2^(-4/3), a positive number. Since n^(1/3) diverges as a series (see p-series), the original series also diverges.
If you have an "exp" function, or a hyperbolic trig function, then write out the functions in terms of e, and you'll see that you do have an n in a power. Try those tests instead.
If you have a logarithm, then try treating it as an extremely small power of n. In fact, for any n^k, with k positive, ln n < n^k for large enough n. So,
which is a converging p-series, so the original series converges as well.
If you have a trigonometric function, check to see if you can find a pattern to the results; this is most likely if you have pi inside the trig function. If you just have a sine or a cosine function, try treating those as if they were constants; that might work, especially with a comparison test (for example, |sin x|<=1).
For example, the sum of sin(pi/2*n) is really 1 + 0 + -1 + 0 + 1 + 0 + -1 + 0 + ... Its sequence of partial sums has no limit, so the series does not converge. The sum of |sin n|/n^2 has terms smaller than 1/n^2, which converges, so the sum of |sin n|/n^2 also converges.
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