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# Find the Maclaurin series for $f(x)$ using the definition of a Maclaurin series. [ Assume that $f$ has a power series expansion. Do not show that $R_n (x) \to 0.$] Also find the associated radius of convergence.$f(x) = \cosh x$

## The Maclaurin series is $\sum_{n=0}^{\infty} \frac{x^{2 n}}{(2 n) !}$which has radius of convergence $=\infty$ .

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find my corn serious for half of ax using the definition of met Lauren Siri's. Okay, So gonna farm. We're going first. Find the primes. The rhythms for that's so zero at zero equals half of eat with zero. Plus, it was not a zero, which is zero, and it goes to one, and a crime at zero is gonna be That's six zero is going to zero and so on were going to find that so forty even order of the derivative at zero is gonna be zero. And for the old number for the other of us is going to one. So we plug in this data into over definition of McLaren, Siri's, and we're going to have This is our definition, and we know that for your r. O. C. It is the affinity because it converts for the whole realign end. With plugging this data, it becomes one purse. So for the even, Order of threw him is just a one and for all, Member, it's just a zero. So Okay, it wasn't this I got So which is equals to K equals two in King. Sorry, Kay. From zero to infinity, Extra pole of two in over two K to Victoria. Yeah, that's all the answer

University of Illinois at Urbana-Champaign

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