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 $$\sum_{n=1}^{\infty}\left(\frac{1}{n+1}-\frac{1}{n+2}\right)$$

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
$$\sum_{n=1}^{\infty}\left(\frac{1}{n+1}-\frac{1}{n+2}\right)$$

Key Concepts

n-th Term Test

The n-th term test checks the limit of the sequence's term as n approaches infinity. If the limit is not zero, the series must diverge. However, if the limit is zero, the test is inconclusive and further analysis using another test is needed.

Geometric Series Test

The geometric series test applies to series where each term is a constant multiple of the previous term. Convergence is determined by the common ratio; if the absolute value of this ratio is less than one, the series converges, and it diverges when it is one or greater.

p-Series Test

The p-series test examines series that have the form of 1/n^p, where convergence depends on the exponent p. Specifically, the series converges when p is greater than one and diverges when p is less than or equal to one.

Telescoping Series Test

The telescoping series test is used for series whose partial sums simplify due to cancellation of terms. By expressing the series in a way that successive terms cancel each other out, one can often directly determine convergence and sometimes even find the exact sum.

Integral Test

The integral test determines the convergence of a series by comparing it to an improper integral. If the corresponding function is positive, continuous, and decreasing, the series converges or diverges together with the integral, making this a useful tool for series that resemble integrable functions.

Direct Comparison Test

The direct comparison test involves comparing the terms of the series to those of a second series with known convergence properties. If every term of the series under consideration is less than or equal to a term of a known convergent series, then the series converges, whereas if it is greater than or equal to a term of a divergent series, it diverges.

Limit Comparison Test

The limit comparison test examines the behavior of the series by comparing the ratio of its terms with the terms of another series with known behavior. If this ratio converges to a positive finite number, both series will either converge or diverge together, making the test particularly useful when the direct comparison is difficult to apply.

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