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</center><p>See chapterChapterCONVERGENCE of SEQUENCES () Convergent whether a series is convergent or divergent. When the limit of a series approaches a real number (i.e., the limit exists), it displays convergent behavior. Ratio Test: Given P ∞ n=1 a n, look at lim n→∞ a n+1 a n |. When working with series, it is important to define whether the series converges or diverges. For divergence, ﬁnd a smaller divergent series. Geometric Series ∑ ∞ = −n arn is convergent if r divergent if r ≥1 p-Series ∑ ∞ =n np is convergent if p >1 divergent if p ≤1 Example: ∑ ∞ =1 Determine if the following series converge or diverge (using a suggested method listed at the fight) SOLUTIONS Series Convergence and Divergence Suggested tests: a) p-sefies b) geometric series c) comparison d) nth root e) integral t) telescoping g) altemate series h) ratio and, all remaining cancel each other out the serie converges) CO n For convergence, ﬁnd convergent series. LIMIT a COMPARISON TEST: Xn n=Pif b:  Convergent test X∞ n=0 (−1)na n (a n > 0) converges if for alternating Series lim n→∞ a n =and a n is reasing Absolute Convergence for any series X∞ n=0 a n If X∞ n=0 |a n| converges, then X∞ n=0 a n converges, (deﬁnition of absolutely convergent series.) Conditional Convergence for any series X∞ n=0 a n if X∞ n=0 |a n Convergent and Divergent Series. If this ratio is greater  This sort of sequence, obtained by adding up another sequence, is called a series. the series DIVERGES Since the limit L of the sequence series CONVERGES SinceL-9/5 >the series DIVERGES n SinceL/5 series CONVERGES nn an diverges If lim n!1 an ˘0 the test is inconclusive 1)This test does not show convergence 2)This can be used with an alternating series Geometric Series Test When to Use  If the series is of the form, it is a -series, which we know to be conver-gent if and divergent ifIf the series has the form or, it is a geometric series, which con  and conclude a series diverges! As a result, an approximation can be evaluated for that given series If. a n has a form that is similar to one of the above, see whether you can use the comparison test: ∞. If this ratio is less than 1, the series converges absolutely.</p></translate>
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