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I need help with these problems please. Problem 1. Check (e^a)(e^b) = (e^(a+b)) by expanding both...

I need help with these problems please.

Problem 1.

Check

(e^a)(e^b) = (e^(a+b))

by expanding both sides in double series in a and b up to fourth order as follows. Make a substitution aat and bbt and expand both sides in t up to (including) t4.

Problem 2.

Calculate the following sum, if convergent.

1624qGTaHyiAAAAAElFTkSuQmCC

Problem 3.

By definition, cosh(x) = (ex + ex)/2 and sinh(x) = (exex)/2. Find series expansion for cosh(x) and sinh(x) expanding near x = 0 up to x6. Calculate

r8JVqI9ApdMSW5GN0H5s2oyyp0d0T3KCeWyl118J


sinh2(x) − cosh2(x)

up to x6. Guess the full answer and prove it.

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