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4. For each of the following series, determine whether it's absolutely convergent, conditionally convergent or divergent. ∞ η-1/3 (Σ n=0 ∞ «Σ n=0 ∞ (i) η=1 η η η + 4n η2 + 6η ∞ ΟΣ n=0 ∞ 1 Σ απόλη η=2 ∞ (1) η=1 η2 2η2 + 1 sin² n 1 + 2n (c) ∞ Σ (k) n=2 ∞ «Σ n=2 Μ8 η=1 n√n-n η η3 – 1 e¹/n η Ο Σε1π. n=0 ∞ 3n+1 4n+2 (m) Σ Σsin(7) η=1 ∞ η=1 3η 2n +4n

4. For each of the following series, determine whether it's absolutely convergent, conditionally convergent or divergent. ∞ η-1/3 (Σ n=0 ∞ «Σ n=0 ∞ (i) η=1 η η η + 4n η2 + 6η ∞ ΟΣ n=0 ∞ 1 Σ απόλη η=2 ∞ (1) η=1 η2 2η2 + 1 sin² n 1 + 2n (c) ∞ Σ (k) n=2 ∞ «Σ n=2 Μ8 η=1 n√n-n η η3 – 1 e¹/n η Ο Σε1π. n=0 ∞ 3n+1 4n+2 (m) Σ Σsin(7) η=1 ∞ η=1 3η 2n +4n

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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4. For each of the following series, determine whether it's absolutely convergent, conditionally convergent, or divergent. (a) \(\sum_{n=0}^{\infty} n^{-1/3}\) (b) \(\sum_{n=0}^{\infty} \frac{n^2}{2n^2 + 1}\) (c) \(\sum_{n=2}^{\infty} \frac{1}{n\sqrt{n} - n}\) (d) \(\sum_{n=0}^{\infty} (-1)^n \cdot \frac{3^{n+1}}{4^{n+2}}\) (e) \(\sum_{n=0}^{\infty} \frac{n}{e^n}\) (f) \(\sum_{n=2}^{\infty} \frac{1}{(\ln n)^n}\) (g) \(\sum_{n=2}^{\infty} \frac{n}{n^3 - 1}\) (h) \(\sum_{n=1}^{\infty} \sin\left(\frac{1}{n}\right)\) (i) \(\sum_{n=1}^{\infty} \frac{n + 4^n}{n^2 + 6^n}\) (j) \(\sum_{n=1}^{\infty} \frac{\sin^2 n}{1 + 2^n}\) (k) \(\sum_{n=1}^{\infty} \frac{e^{1/n}}{n}\) (l) \(\sum_{n=1}^{\infty} \frac{3^n}{2^n + 4^n}\)
Transcribed Image Text:4. For each of the following series, determine whether it's absolutely convergent, conditionally convergent, or divergent. (a) \(\sum_{n=0}^{\infty} n^{-1/3}\) (b) \(\sum_{n=0}^{\infty} \frac{n^2}{2n^2 + 1}\) (c) \(\sum_{n=2}^{\infty} \frac{1}{n\sqrt{n} - n}\) (d) \(\sum_{n=0}^{\infty} (-1)^n \cdot \frac{3^{n+1}}{4^{n+2}}\) (e) \(\sum_{n=0}^{\infty} \frac{n}{e^n}\) (f) \(\sum_{n=2}^{\infty} \frac{1}{(\ln n)^n}\) (g) \(\sum_{n=2}^{\infty} \frac{n}{n^3 - 1}\) (h) \(\sum_{n=1}^{\infty} \sin\left(\frac{1}{n}\right)\) (i) \(\sum_{n=1}^{\infty} \frac{n + 4^n}{n^2 + 6^n}\) (j) \(\sum_{n=1}^{\infty} \frac{\sin^2 n}{1 + 2^n}\) (k) \(\sum_{n=1}^{\infty} \frac{e^{1/n}}{n}\) (l) \(\sum_{n=1}^{\infty} \frac{3^n}{2^n + 4^n}\)
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