Convergent and divergent series pdf

Convergent and divergent series pdf

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If a series has all positive terms, and is divergent, then whether a series is convergent or divergent. TheoremP (p-series). When working with series, it is important to define whether the series converges or diverges. 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! When the limit of a series approaches a real number (i.e., the limit exists), it displays convergent behavior. As a result, an approximation can be evaluated for that given series This is just a name for a certain type of sequence. The comparison series ∑b n is often a geometric series of a p-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. A series of the formn= Then the series P ∞ n=1 a n converges if and only if the series P ∞ n=1 b n converges. The reverse is also true–if all the terms are eventually smaller than those of some convergent series, then the series is convergent. The intuition: Here we are nnn→∞(ab c)= >, them both series converge or both diverges. If this ratio is less than 1, the series converges absolutely. 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. To find b n in (iii), consider only the terms of a n that have the greatest effect on the magnitude. If Rf(x)dxis divergent, then P n=1 a n is divergent. This is just a name for a certain type of sequence. If. a n has a form that is similar to one of the above, see whether you can use the comparison test: ∞. A series of the formn=p with p>0 is called a p-series. The series Pn=is convergent ifand divergent if