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1. Determine whether the following series converge. Rigorously justify your answers (using the convergence tests from the lecture). 2n 2n (a) (b) Σ n! n=1 1+ 2n n=0 n=1

1. Determine whether the following series converge. Rigorously justify your answers (using the convergence tests from the lecture). 2n 2n (a) (b) Σ n! n=1 1+ 2n n=0 n=1

Advanced Engineering 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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n=0 \m=0 Safari File Edit View History Bookmarks Window Help 100% 4A Sun 20:03 A learn-eu-central-1-prod-fleet01-xythos.content.blackboardcdn.com The Try Guys Korean FIRE Noodle Challe... Content Content X Bb https://learn-eu-central-1-prod-fleet01... b Answered: H1. Find the solution to the... 1. Determine whether the following series converge. Rigorously justify your answers (using the convergence tests from the lecture). n 2n 2n (a) Σ (b) (c) n² + 1 n=1 п! n=1 1+ 2n' n=0 (f) £(-1)n+1[sin n| n4 n2 1 (d) (e) n4 + 1 n=1 Зп + 2 n=1 n=1 n! 1 Σ (h) ΣΙΣ (2n)! 2m n=1 2. [Python question: submit a print-out of your answer] We have shown that the harmonic series E diverges because its sequence of 1 'n=l n partial sums k 1 Sk = n n=1 is unbounded above. Hence, given any M e R, there exists some k e N such that > M. By experimenting with Python, find the smallest k E N such that Sk a) sk > 4, (b) Sk > 8, (c) Sk > 12, (d) sk > 16. A 4,621 FEB 27 W lil O
Transcribed Image Text:n=0 \m=0 Safari File Edit View History Bookmarks Window Help 100% 4A Sun 20:03 A learn-eu-central-1-prod-fleet01-xythos.content.blackboardcdn.com The Try Guys Korean FIRE Noodle Challe... Content Content X Bb https://learn-eu-central-1-prod-fleet01... b Answered: H1. Find the solution to the... 1. Determine whether the following series converge. Rigorously justify your answers (using the convergence tests from the lecture). n 2n 2n (a) Σ (b) (c) n² + 1 n=1 п! n=1 1+ 2n' n=0 (f) £(-1)n+1[sin n| n4 n2 1 (d) (e) n4 + 1 n=1 Зп + 2 n=1 n=1 n! 1 Σ (h) ΣΙΣ (2n)! 2m n=1 2. [Python question: submit a print-out of your answer] We have shown that the harmonic series E diverges because its sequence of 1 'n=l n partial sums k 1 Sk = n n=1 is unbounded above. Hence, given any M e R, there exists some k e N such that > M. By experimenting with Python, find the smallest k E N such that Sk a) sk > 4, (b) Sk > 8, (c) Sk > 12, (d) sk > 16. A 4,621 FEB 27 W lil O
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