Lecture - 5

1

Test for Convergence

Contents

•Series of Positive terms

•Algebra of sums

•Comparison Test

2

•Let (an) = a1, a2,……an… be a

sequence of real numbers. (ai > 0)

•Notion of sequence of partial sums

follows here too

However, s1= a1

s2 = a1 + a2 ………

Note s1 < s2 < s3 < ……< sn < ……

3

Series of Positive terms

•Hence, we have a monotonically

increasing sequence

•If (sn) is bounded, or unbounded then

will be a convergent series or a

divergent series

•A series of positive term will never

oscillate

4

Series of Positive terms

a1n

n

•If converges to a and

converges to b then

• converges to a + b

• converges to ka

5

Infinite Series – Algebra of sums

a1n

n

b1n

n

)b(a1n

nn

ka1n

n

convergence

Infinite Series of positive terms

Usage of Geometric Series

Algebra of sums

Comparison Test

6

Summary

1. Prove that (an) is convergent if

and only if is

convergent

2. Prove that sum of a convergent

and a divergent series will diverge

3. Test the convergence of

7

Questions

n

1

1n

)a - (a1n

n1n

n

1

1n2