Lecture - 4

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Convergence of Infinite Series

Contents

• Infinite Series - Review•Specific Series•Convergence

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Let (an) = a1, a2,……an… be a sequence of real numbers.

The expression

=a1+ a2 +…+ an +…….

is called an infinite series of real numbers

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Series of Positive terms

∑∑∞

=n

1nn

aor a

•If sn= a1+a2+……+an then (sn) is called the sequence of partial sums of the given series.

•The series is said to Converge,

Diverge or Oscillate according as the sequence (sn) Converges, Diverges or Oscillates

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Series of Positive terms

a1n

n∑∞

=

•The Geometric series

1+ r + r2 +…+ rn ……

sn= 1+ r + r2 +…+ rn =

•(rn) → 0 when 0 ≤ r < 1

•(rn) → ∞ when r >1

•When r =1 (sn)= (n) → ∞5

Infinite Series - Example

r1

r1 n

−−

•The Geometric series – Cont’dsn= 1+ r + r2 +…+ rn =

•r = -1 then sn is 0 or 1 (n: even / odd)

(sn) oscillates finitely

•When r < -1 (rn) oscillates infinitely 6

Infinite Series - Example

r1

r1 n

−−

• Let be a convergent series

converging to the sum s. Then – Converse: need not imply the convergence of the series

– However, if then the series will not converge

• The case would provide strict conclusion for series of +ve terms

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Infinite Series - Convergence a

1nn∑

∞

=0a lim

nn=

∞→

0a limnn

=∞→

0a limnn

≠∞→

(an) in R is convergent ⇔ (an) is CauchyUseful in convergence Infinite SeriesSequence of partial sums define the convergence of an infinite series

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Summary

? Try CCC to study the convergence of?(1/n), ((-1)n), (n)

? Test the convergence of 1 + 2 + …….

? Test the convergence of

? Test the convergence of -1 + 2 - 3+…. (Use CCC)

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Questions

2

11n

n∑∞

=

? Try CCC to study the convergence of?(1/n), ((-1)n), (n)

? Test the convergence of 1 + 2 + …….

? Test the convergence of

? Test the convergence of -1 + 2 - 3+…. (Use CCC)

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Questions

2

11n

n∑∞

=