An infinite series is the sum of an infinite sequence. More precisely, the sum of an infinite series is defined as the limit of the sequence of the partial sums of the terms of the series, provided this limit exists. If no finite limit exists, then we say that the series is divergent.
Infinite series can also be viewed as the extension of the algebraic operation of addition to infinitely many terms. Convergence means that for the given series this extension is consistent with, and can be approximated by, the conventional notion of finite sums.
In class you learned various tests that can help to decide if a series is convergent or divergent. But with the exception of the geometric series, no general formulas are known for the actual sum of a convergent series. The real value of the tests is that once the convergence has been proved, the sum can be approximated numerically. When using approximations, it is important to know how accurate they are. Fortunately, there are several formulas which allow to estimate the approximation error and to determine the number of terms that need to be added up in order to approximate the sum of an infinite series with the required accuracy. The best known error formulas are the following.
a.)Alternating series reminder estimate:
Applies to series with alternating signs and terms decreasing in absolute value .
b.)Integral test reminder estimate:
Applies to series with positive terms whose convergence can be proved by the integral tests.