Problem 1. For this problem, recall the following Maclaurin series, valid for all x E R:cos(x) = D(2n)!(-1)"Σ(-1)"sin(x) =„2n+1n!n=0(2n + 1)!"n=0n=0It turns out these series are not only valid for all real numbers x; they are also valid for all complex numbersz = x + iy E C, where i = v-1.(a) Calculate i2, i³, and it in terms only of ±i and ±1. In general, what is i2n and i2n+1? (Hint: Your answer tothese questions will involve a (-1)"-term.)(b) Use the above Maclaurin series to show e2® = cos(0) + i sin(0).(c) Prove Euler's identity: e?™ +1= 0. (Many consider this the most beautiful equation in all of mathematics.)

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Problem 1. For this problem, recall the following Maclaurin series, valid for all x E R: x" cos(x) = E (-1)" (-1)" et n!' n=0 (2n)! n=0 sin(x) (2n + 1)!" 72n+1 n=0 It turns out these series are not only valid for all real numbers x; they are also valid for all complex numbers z = x + iy E C, where i = V-1. (a) Calculate i?, i3, and i4 in terms only of ±i and ±1. In general, what is i2n and i2n+1? (Hint: Your answer to these questions will involve a (-1)"-term.) (b) Use the above Maclaurin series to show eil = cos(0) + i sin(0). (c) Prove Euler's identity: en +1 = 0. (Many consider this the most beautiful equation in all of mathematics.)

Problem 1. For this problem, recall the following Maclaurin series, valid for all x E R: x" cos(x) = E (-1)" (-1)" et n!' n=0 (2n)! n=0 sin(x) (2n + 1)!" 72n+1 n=0 It turns out these series are not only valid for all real numbers x; they are also valid for all complex numbers z = x + iy E C, where i = V-1. (a) Calculate i?, i3, and i4 in terms only of ±i and ±1. In general, what is i2n and i2n+1? (Hint: Your answer to these questions will involve a (-1)"-term.) (b) Use the above Maclaurin series to show eil = cos(0) + i sin(0). (c) Prove Euler's identity: en +1 = 0. (Many consider this the most beautiful equation in all of mathematics.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Problem 1. For this problem, recall the following Maclaurin series, valid for all x E R:
cos(x) = D
(2n)!
(-1)"
Σ
(-1)"
sin(x) =
„2n+1
n!
n=0
(2n + 1)!"
n=0
n=0
It turns out these series are not only valid for all real numbers x; they are also valid for all complex numbers
z = x + iy E C, where i = v-1.
(a) Calculate i2, i³, and it in terms only of ±i and ±1. In general, what is i2n and i2n+1? (Hint: Your answer to
these questions will involve a (-1)"-term.)
(b) Use the above Maclaurin series to show e2® = cos(0) + i sin(0).
(c) Prove Euler's identity: e?™ +1= 0. (Many consider this the most beautiful equation in all of mathematics.)

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