- #1

frozenguy

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## Homework Statement

(a)Use Definition 10.8.1 to find the Maclaurin series for f(x) = sinh x. Express your answer using Σ notation.

(b) Find the interval of convergence for the series found in part (a).

(c) Use the Remainder Theorems 10.7.4 and 10.9.2 to show that the series found in part (a) converges to f(x) = sinh x on the interval of convergence found in part (b).

## The Attempt at a Solution

So I got a) and b) I'm sure, but I'm having trouble with c).

For (a) I got [tex]\sum\frac{x^{2k+1}}{(2k+1)!}[/tex] 0-->inf

for (b) I used the abs ratio test to find the limit = 0 therefore the interval is [tex](-\infty,+\infty)[/tex]

10.7 is mac/taylor polys

10.8 is mac/taylor series (power series)

10.9 is convergence of taylor series.

The remainder theorm of 10.7 states that "If the function f can be differentiated n+1 times on an interval I containing the number Xo, and if M is an upper bound for [tex]\left|f^{(n+1)}(x)\right|[/tex] on I,

that is, [tex]\left|f^{(n+1)}(x)\right|\leq M[/tex]

for all x in I, then [tex]\left|R_{n}(x)\right|\leq\frac{M}{(n+1)!}\left|x-x_{0}\right|^{n+1}[/tex]

But I don't understand how to use that theorem or 10.9.2 that says The equality: [tex]f(x)=\sum\frac{f^{(k)}(x_{0}}{k!}(x-x_{0})^k[/tex]

holds at a point x if and only if lim of Rn(x)=0 as n approaches +inf.

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