Exercise 2.6.10: a) Prove that lim n¹/n = 1 using the following procedure: Write n¹/n = 1+ bn and note bn > 0. Then show that (1+b,)” ≥ "(n=¹) b² and use this to show that lim bn n(n-1) = 0. 2 b) Use the result of part a) to show that if anx" is a convergent power series with radius of convergence R, then nanx" is also convergent with the same radius of convergence.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Exercise 2.6.10:
a) Prove that lim n¹/n = 1 using the following procedure: Write n¹/n = 1+ bn and note bn > 0. Then show
that (1+b,)” ≥ "(n=¹) b² and use this to show that lim bn
n(n-1)
= 0.
2
b) Use the result of part a) to show that if Σanx" is a convergent power series with radius of convergence R,
then nanx" is also convergent with the same radius of convergence.
Transcribed Image Text:Exercise 2.6.10: a) Prove that lim n¹/n = 1 using the following procedure: Write n¹/n = 1+ bn and note bn > 0. Then show that (1+b,)” ≥ "(n=¹) b² and use this to show that lim bn n(n-1) = 0. 2 b) Use the result of part a) to show that if Σanx" is a convergent power series with radius of convergence R, then nanx" is also convergent with the same radius of convergence.
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