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Name Math 106 November 7, 2003 Exam 2
โ^ โ n=
(โ1)n^ (2n^ + 1)
3 (2n + 1)^4 + 4 =^
converge, or does it diverge? Explain. If it converges, you will receive extra credit if you can find what number it converges to.
(a)
n=
n^2 + 1 n^4 + 1 (b)
n=
n^3 + 1 n^4 + 1
at 0. For what values of y does the series converge? How do you know?
(ii) Write down the first four nonzero terms, and if possible all the terms, of the Taylor series of (^) (1 โy y (^2) ) 2
at 0.
These functions are defined by cosh x = e
x (^) + eโx 2 and^ sinh^ x^ =^
ex^ โ eโx 2 respectively.
Explain why cosh x =
n=
x^2 n (2n)! and^ sinh^ x^ =
n=
x^2 n+ (2n + 1)!.
(d) Write down the Taylor series of cos x and sin x at 0.
(e) Using your answers above, or otherwise, try to figure out what functions the following series converge to:
(i)
n=
x^4 n (4n)! (ii)
n=
x^4 n+ (4n + 1)! (iii)
n=
x^4 n+ (4n + 2)! (iv)
n=
x^4 n+ (4n + 3)!
