University Mathematics Handbook (2015)
VI. Series
Chapter 1. Basic Concepts
1.1 Series of Numbers. Summation of Series
a. Let be a sequence of numbers.
The expression is called infinite series or just a series, and ak is the general term of the series.
The terms of the series add up to partial sums:
The result is a sequence of partial sums .
b. Series is convergent if there exists a finite limit of the sequence of partial sums . is called the sum of the series. Written .
If the limit of does not exist, or is infinity, the series is called divergent.
c. Examples:
1. Geometric Series
converges if and only if and its sum is .
2. Leibniz Series
3. Series is divergent.
4. Harmonic Series .
1.2 Series Remainder
The series is the -th remainder of the series , .
1.3 Telescoping Series
a. Let be a sequence of numbers. Series is called telescoping series.
b. A telescoping series is convergent if and only if sequence is convergent.
c. If sequence converges to , then the sum of the telescoping series is .
1.4 Properties of Convergent Series
a. Cauchy's Criterion: Series converges if and only if
, , ,
b. Necessary Condition of Convergence
If series converges, then .
This is an insufficient condition: For instance, for harmonic series , there holds , yet the series is divergent.
c. If series is convergent, then the sequence of remainders converges to zero. That is, for every there exists such that for all , there holds .
1.5 Operation on Series
a. Removing a finite number of terms from a series, or adding a finite number of terms to it, does not affect the convergence or divergence of the series.
Attention: It does change the sum of the series.
b. If converges to , then, for every constant the series converges to .
If series is divergent, then, for all , the series , is divergent.
c. If series , are convergent, then, the series is convergent, and .