Here is a short review of numerical series which you may find helpful.
REVIEW OF NUMERICAL SERIES
SEQUENCES
A sequence is a list of real numbers. It is called convergent if it has a limit. An increasing sequence has a limit when it has an upper bound. SERIES
(Geometric series,rational numbers as decimals, harmonic series,divergence test)
Given numbers forming a sequence let us define the nth partial sum as sum of the first n of them
The SERIES is convergent if the SEQUENCE is. In other words it converges if the partial sums of the series approach a limit.
A necessary condition for the convergence of this SERIES is that a's have limit 0. If this fails, the series diverges.
The harmonic series 1+(1/2)+(1/3)+... diverges.
This illustrates that the terms having limit zero does not guarantee the convergence of a series.
A series with positive terms ,i.e. for all n, converges
exactly when its partial sums have an upper bound.
The geometric series converges exactly when
INTEGRAL AND COMPARISON TESTS
(Integral test,p-series, comparison tests for convergence and divergence, limit comparison test)
Integral test: Suppose is positive and DECREASING for all large enough x. Then the following are equivalent:
I. is finite, i.e. converges.
S. is finite, i.e. converges.
This gives the p - test: converges exactly when
Comparison test: Suppose there is a fixed number K such that
for all sufficiently large n:
Convergence. If converges then so does
Divergence. If diverges then so does .
(Positive series having smaller terms are more likely to converge.)
Limit comparison test: SUPPOSE: , and
exists. Moreover, R is not zero.
THEN and
both converge or both diverge.
OTHER CONVERGENCE TESTS FOR SERIES
(Alternating series test, absolute convergence, RATIO TEST)
Alternating series test: Suppose the sequence is decreasing and has limit zero. Then converges.
This applies to (1)-(1/2)+(1/3)-(1/4)+...
Absolute Convergence Test: IF converges,
THEN converges.
Ratio test:
SUPPOSE has limit equal to r.
IF then CONVERGES.
IF the DIVERGES.