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 $$ \sum_{n=1}^{\infty} \frac{\sqrt{n}}{n} $$

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
$$
\sum_{n=1}^{\infty} \frac{\sqrt{n}}{n}
$$

Key Concepts

n-th Term Test

The n-th Term Test for divergence states that if the limit of the sequence of terms in a series does not approach zero, then the series diverges. It is often the first test applied because a nonzero limit immediately indicates divergence. However, if the limit is zero, the test is inconclusive, and further tests must be applied to determine convergence or divergence.

Geometric Series Test

The Geometric Series Test applies specifically to series in which each term is a constant multiple of the previous term, forming a geometric progression. A geometric series converges if the absolute value of the common ratio is less than one; otherwise, it diverges. This test is a powerful tool for series that fit this pattern, but it is not applicable to series whose terms do not have a constant ratio.

p-Series Test

The p-Series Test is used for series of the form ?1/n^p. The test states that the series converges if p > 1 and diverges if p ? 1. This is a fundamental test in the study of infinite series as it provides a clear criterion for convergence based solely on the exponent of n in the denominator.

Telescoping Series Test

The Telescoping Series Test applies to series where many terms cancel out when partial sums are written out, leading to a simplified expression. This cancellation pattern makes it easier to evaluate the limit of the partial sums, thereby determining convergence or divergence. The concept is particularly useful when the series can be expressed as the difference of successive terms.

Integral Test

The Integral Test involves comparing a series to an improper integral. If the function corresponding to the terms of the series is continuous, positive, and decreasing, then the convergence of the series is equivalent to the convergence of the integral of that function over an appropriate interval. This test can be an effective way to handle series that do not easily fit into other standard forms.

Direct Comparison Test

The Direct Comparison Test is used to determine convergence or divergence by comparing the series in question with another series whose convergence properties are known. If each term of the series is less than or equal to a corresponding term of a known convergent series, then the series converges; conversely, if each term is greater than or equal to a term of a known divergent series, then the series diverges.

Limit Comparison Test

The Limit Comparison Test involves taking the limit of the ratio of the general term of the series in question to the general term of another series with known convergence behavior. If the limit exists and is a positive finite number, both series either converge or diverge together. This test is particularly useful when the series do not directly compare term by term but have similar asymptotic behavior.

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