Find the radius of convergence of the power series (1/2)z+(1.2/2.5)z²+(1.3.5/2.5.8)z³+.....

Описание: Complex Analysis Theorem from Power Series
Statement/Theorem /Prove that :-

Find the radius of convergence of the power series

(1/2)z+(1.2/2.5)z²+(1.3.5/2.5.8)z³.....

Solution/ Proof:

The coefficient of zⁿ
aₙ={1.3.5.....(2n-1)}/{2.5.8.....(3n-1)}
then
aₙ₊₁ ={1.3.5.....(2n-1)(2n+1)}/{2.5.8.....(3n-1)(3n+2)}
The radius of convergence

(1/R)=limₙ→∞|aₙ₊₁/aₙ|
⇒(1/R)=limₙ→∞|(2n+1)/(3n+2)|
⇒(1/R)=limₙ→∞|n{2+(1/n)}/n{3+(2/n)}|
⇒(1/R)=(2+0/3+0)
⇒(1/R)=2/3
--------------------------
⇒R=3/2
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This is required radius of convergence .
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Solved..!!!
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#complexanalysis #csirnet #powerseries #radiusofconvergence
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