TABLE OF CONTENTS In this chapter we introduce infinite series and prove some basic convergence theorems. We also introduce power series a very powerful tool in analysis.
Study of the behaviour of the terms of a sequence when they are successively added leads to infinite series. For an arbitrary sequence n 
a n ? 
we can form another sequence by successive additions as follows:
| (11.1) |  |
To indicate that we consider n s n rather than n a n we write
| (11.2) |  |
The symbol (11.2) is just an abbreviation for the sequence n s n, with s n as in (11.1). We shall call (11.2) a series or an infinite series ; a n is the n th term of (11.2); s n is the n th partial sum of (11.2).
It is usually clear from the context what the partial sums for a series like (11.2) are. On the other hand, if 
for some positive integer k and every natural number i then
However the symbol ? k ?i is ambiguous. On occasions like this we would write 
rather than ? k ? i. The letter i in (11.2) can be replaced by another letter without altering the meaning of the symbol; for example we can write ? a j or ? a k instead of ? a i. For instance, the geometric series, that is the sequence n s n
