If some of the terms are negative, then look at the series of absolute values instead. If that converges, then the original will as well, because of the Absolute Convergence Test. Try to get rid of negative exponents, and treat them as fractions instead. If you have fractions inside of fractions, try finding common denominators and reduce to just one fraction.

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.

For instance,

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.

Copyright © 1996 Department of Mathematics, Oregon State University

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