Test for convergence or divergence, using each test at least...

Test for convergence or divergence, using each test at least once. Identify which test was used. (a) $n$ th-Term Test (b) Geometric Series Test (c) $p$ -Series Test (d) Telescoping Series Test (e) Integral Test (f) Direct Comparison Test (g) Limit Comparison Test $$\sum_{n=1}^{\infty} \frac{n}{\left(n^{2}+1\right)^{2}}$$

Test for convergence or divergence, using each test at least once. Identify which test was used.
(a) $n$ th-Term Test
(b) Geometric Series Test
(c) $p$ -Series Test
(d) Telescoping Series Test
(e) Integral Test
(f) Direct Comparison Test
(g) Limit Comparison Test
$$\sum_{n=1}^{\infty} \frac{n}{\left(n^{2}+1\right)^{2}}$$

Key Concepts

nth-Term Test

The nth-Term Test for Divergence is used to determine if a series might diverge by checking whether the limit of its individual terms as n approaches infinity is nonzero. If the limit is not zero (or does not exist), the series diverges, making this a useful first step in series analysis.

Geometric Series Test

The Geometric Series Test applies to series whose terms form a geometric progression, that is, each term is a constant multiple (common ratio) of the previous term. A geometric series converges if and only if the absolute value of the common ratio is less than one, otherwise it diverges.

p-Series Test

The p-Series Test is used for series of the form ?1/n^p. The convergence of a p-series is determined solely by the exponent p; the series converges if p > 1 and diverges if p ? 1, making it a benchmark for comparing other series.

Telescoping Series Test

The Telescoping Series Test is applied when a series has terms that cancel in a successive manner. By expressing the partial sums in a simplified form, many terms cancel out, which allows for an easier analysis of convergence by examining the behavior of the remaining terms.

Integral Test

The Integral Test links the convergence of a series to the convergence of an improper integral. When a series' general term is positive, continuous, and decreasing, integrating the corresponding function can provide direct insight into whether the series converges or diverges.

Direct Comparison Test

The Direct Comparison Test involves comparing the terms of a given series to those of another series with known convergence properties. If every term of the original series is less than or equal to the corresponding term in a convergent series (or greater than or equal to those of a divergent series), conclusions about convergence can be made.

Limit Comparison Test

The Limit Comparison Test is used when direct term-by-term comparison is not straightforward. By taking the limit of the ratio of the series' term to the corresponding term of a known series, if the limit is a positive finite number, both series will either converge or diverge together.

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