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The radius of convergence of the power series 
- a)1/4
- b)1
- c)2
- d)4
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
The radius of convergence of the power series a)1/4b)1c)2d)4Correct an...
Recall from the Power Series page that we saw that a power series will converge at it's center of convergence c, and that it is possible that a power series can converge for all x∈R or on some interval centered at the center of convergence. If a power series converges on some interval centered at the center of convergence, then the distance from the center of convergence to either endpoint of that interval is known as the radius of convergence which we more precisely define below.
Example
Determine the radius of convergence of the power series ∑∞n=011+n3(x+6)n.
We note that the center of convergence is c=−6. Now we want to find the radius of convergence using the ratio test. Let an=11+n3. Thus:
L=limn→∞∣∣∣an+1an∣∣∣=limn→∞an+1an=limn→∞11+(n+1)311+n3=limn→∞1+n31+(n+1)3=limn→∞1+n32+3n+3n2+n3=limn→∞1n3+12n3+3n2+3n+1=1
So L=1, and so the radius of convergence is R=1L=1.
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The radius of convergence of the power series a)1/4b)1c)2d)4Correct an...
The corect answer is 1/4 option: (A). By using Radius of convergence formula. lt n->
∞|an+1/an| =1/R. Where R is RADIUS of convergence.
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