Cn Cn+1 ists (or if the ratio properly diverges to ∞, in which case we say that R = ∞), then the power series R = lim n→∞ Σ n=0 the complex variable w converges absolutely for [w] < R and does not converge for |w| > R. We call R he radius of convergence of the power series. En wr 1. Several weeks ago, we have given two definitions of the complex exponential. One of them was d dz ∞ e² = Σk!* k=0 (a) Compute the radius of convergence of this series. (b) Check explicitly that the derivative and summation can be exchanged, i.e. = d dz 8 k=0 k=0 dzk dz k!' by computing the complex derivative of each term in the series and reindexing this sum of deriva- tives.
Cn Cn+1 ists (or if the ratio properly diverges to ∞, in which case we say that R = ∞), then the power series R = lim n→∞ Σ n=0 the complex variable w converges absolutely for [w] < R and does not converge for |w| > R. We call R he radius of convergence of the power series. En wr 1. Several weeks ago, we have given two definitions of the complex exponential. One of them was d dz ∞ e² = Σk!* k=0 (a) Compute the radius of convergence of this series. (b) Check explicitly that the derivative and summation can be exchanged, i.e. = d dz 8 k=0 k=0 dzk dz k!' by computing the complex derivative of each term in the series and reindexing this sum of deriva- tives.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Let us recall from recitation (or Section V.3 of [G]) that if the sequence (cn)o of complex coefficients
is such that the limit
Cn
R = lim
n→∞ Cn+1
exists (or if the ratio properly diverges to ∞, in which case we say that R = ∞), then the power series
n=0
in the complex variable w converges absolutely for |w| < R and does not converge for |w| > R. We call R
the radius of convergence of the power series.
1. Several weeks ago, we have given two definitions of the complex exponential. One of them was
d
dz
GnWn
-e²
e² =
(a) Compute the radius of convergence of this series.
(b) Check explicitly that the derivative and summation can be exchanged, i.e.
=
k=0
zk
k!
k
d
«Σ-Σ
=
dz k!
k=0
∞
dzk
dz k!'
k=0
by computing the complex derivative of each term in the series and reindexing this sum of deriva-
tives.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a04a0c8-a6a9-473e-b4ed-0840bde177a1%2Fb4071183-15a0-4d9d-8f8f-9f1ec422787e%2Fky2ix6_processed.png&w=3840&q=75)
Transcribed Image Text:Let us recall from recitation (or Section V.3 of [G]) that if the sequence (cn)o of complex coefficients
is such that the limit
Cn
R = lim
n→∞ Cn+1
exists (or if the ratio properly diverges to ∞, in which case we say that R = ∞), then the power series
n=0
in the complex variable w converges absolutely for |w| < R and does not converge for |w| > R. We call R
the radius of convergence of the power series.
1. Several weeks ago, we have given two definitions of the complex exponential. One of them was
d
dz
GnWn
-e²
e² =
(a) Compute the radius of convergence of this series.
(b) Check explicitly that the derivative and summation can be exchanged, i.e.
=
k=0
zk
k!
k
d
«Σ-Σ
=
dz k!
k=0
∞
dzk
dz k!'
k=0
by computing the complex derivative of each term in the series and reindexing this sum of deriva-
tives.
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