Let x0, x1,x2,... be the sequence such that xo = 1 and for n ≥ 0, Xn+1 = ln(en — Xn) (as usual, the function In is the natural logarithm). Show that the infinite series xo + x1 + x₂ + ·· ... converges and find its sum.
Let x0, x1,x2,... be the sequence such that xo = 1 and for n ≥ 0, Xn+1 = ln(en — Xn) (as usual, the function In is the natural logarithm). Show that the infinite series xo + x1 + x₂ + ·· ... converges and find its sum.
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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![Let \( x_0, x_1, x_2, \ldots \) be the sequence such that \( x_0 = 1 \) and for \( n \geq 0 \),
\[
x_{n+1} = \ln(e^{x_n} - x_n)
\]
(as usual, the function \(\ln\) is the natural logarithm). Show that the infinite series \( x_0 + x_1 + x_2 + \cdots \) converges and find its sum.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1d5e88e6-b1af-4aea-9b08-2dadd85f5e2c%2F3d375432-b5d7-4290-895a-353e843e1088%2Fcdibk0p_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( x_0, x_1, x_2, \ldots \) be the sequence such that \( x_0 = 1 \) and for \( n \geq 0 \),
\[
x_{n+1} = \ln(e^{x_n} - x_n)
\]
(as usual, the function \(\ln\) is the natural logarithm). Show that the infinite series \( x_0 + x_1 + x_2 + \cdots \) converges and find its sum.
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