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## 11.3

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**The Integral Test**11.3 • The Comparison Tests • 11.4 • 11.5 • Alternating Series • 11.6 • Ratio and Root Tests**Definition of Convergence for an infinite series:**• Let be an infinite series of positive terms. • The series converges if and only if the sequence of partial sums, , converges. This means: Divergence Test: • If , the series diverges. • is divergent since • Example: The series • However, • does not imply convergence!**Geometric Series:**The Geometric Series: converges for If the series converges, the sum of the series is: Example: The series with and converges . The sum of the series is 35.**p-Series:**The Series: (called a p-series) converges for and diverges for Example: The series is convergent. The series is divergent.**Integral test:**If f is a continuous, positive, decreasing function on with then the series converges if and only if the improper integral converges. Example: Try the series: Note: in general for a series of the form:**Comparison test:**If the series and are two series with positive terms, then: • If is convergent and for all n, then converges. • If is divergent and for all n, then diverges. • (smaller than convergent is convergent) • (bigger than divergent is divergent) Examples: which is a divergent harmonic series. Since the original series is larger by comparison, it is divergent. which is a convergent p-series. Since the original series is smaller by comparison, it is convergent.**Limit Comparison test:**If the the series and are two series with positive terms, and if where then either both series converge or both series diverge. Useful trick: To obtain a series for comparison, omit lower order terms in the numerator and the denominator and then simplify. Examples: For the series compare to which is a convergent p-series. For the series compare to which is a divergent geometric series.**Alternating Series test:**If the alternating series satisfies: and then the series converges. Definition: Absolute convergence means that the series converges without alternating (all signs and terms are positive). Example: The series is convergent but not absolutely convergent. Alternating p-series converges for p > 0. Example: The series and the Alternating Harmonic series are convergent.**Ratio test:**• If then the series converges; • If the series diverges. • Otherwise, you must use a different test for convergence. If this limit is 1, the test is inconclusive and a different test is required. Specifically, the Ratio Test does not work for p-series. Example:**11.7**• Strategy for Testing Series**Summary:**• Apply the following steps when testing for convergence: • Does the nth term approach zero as n approaches infinity? If not, the Divergence Test implies the series diverges. • Is the series one of the special types - geometric, telescoping, p-series, alternating series? • Can the integral test, ratio test, or root test be applied? • Can the series be compared in a useful way to one of the special types?**Example**• Determine whether the series converges or diverges using the Integral Test. • Solution: Integral Test: Since this improper integral is divergent, the series (ln n)/n is also divergent by the Integral Test.**Example 1**• Since as an ≠ as n , we should use the Test for Divergence.**Example 2**Since an is an algebraic function of, we compare the given series with a p-series. The comparison series for the Limit Comparison Test is where**Example 4**Since the series is alternating, we use the Alternating Series Test.**Example 5**• Since the series involves k!, we use the Ratio Test.**Example 6**Since the series is closely related to the geometric series , we use the Comparison Test.