Convergence of SeriesWm C BauldryBoone, NCSeries of Real NumbersA series is generated by summing a sequence of terms. The partial sums LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn of the series are given by 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. We say the series converges to a limit when the sequence of partial sums converges. Since it's very difficult to tell by inspection when a series converges, we select from a set of convergence tests that increase in power, but also increase in the effort needed for their computation. We must interpret the results of any test to determine what it tells us about the series in question.The Series Test Calculator let's us enter the general term LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn, the beginning index LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPUY+, and then have Maple perform the computations by clicking the Do the Tests button. After interpreting the results, then click the Evaluate button to see Maple's answer with a graph of the partial sums LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn and the terms LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn. Compare your results with Maple's.Try the Series Test Calculator with series that you know; for example, try the calculator on series with terms given by 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 and 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 and n0 :=. Now, click to: use DocumentTools in a := unapply(MathML[ImportModified](GetProperty('MCa','value')),n); n0:= eval(MathML[ImportModified](GetProperty('MCn','value'))); SetProperty('MCRatio','value', Limit(a(n+1)/a(n),n=infinity) =limit(a(n+1)/a(n),n=infinity)); SetProperty('MCRoot','value', Limit(a(n)^(1/n),n=infinity) =limit(a(n)^(1/n),n=infinity)); SetProperty('MCRaabe','value', Limit(n*(1-a(n+1)/a(n)),n=infinity) =limit(n*(1-a(n+1)/a(n)),n=infinity)); SetProperty( 'MCIntegral','value', Int(a(n),n=n0..infinity) =int(a(n),n=n0..infinity)); SetProperty('Button2','enabled',true); SetProperty('MCResult','value', Sum(a(n),n=n0..infinity)= `?`); Do(%Plot0 = plot(0,n=0..20,color=black)); end use; D'Alembert's Ratio Test: Cauchy's Root Test: Raabe's Test: The Integral Test: Maple's Results The Series: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GMDYmRjJGNUY4L0Y7USVhdXRvRidGKy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== use DocumentTools in a := unapply(MathML[ImportModified](GetProperty('MCa','value')),n); n0:= eval(MathML[ImportModified](GetProperty('MCn','value'))); SetProperty('MCResult','value', Sum(a(n),n=n0..infinity)=sum(a(n),n=n0..infinity)); SetProperty('Button2','enabled',false); Do(%Plot0 = plot([[seq([n,sum(a(j),j=n0..n)],n=n0..20)],[seq([n,a(n)],n=n0..20)]],style=point, symbol=[solidbox,solidcircle], legend=[Series,Terms])); end use; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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NiI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFHuse DocumentTools in a := 'a': n0 := 'n0': SetProperty('MCa','value',evaln(a)[n]); SetProperty('MCn','value',evaln(n)[0]); SetProperty('MCRatio','value', Limit(a(n+1)/a(n),n=infinity) ); SetProperty('MCRoot','value', Limit(a(n)^(1/n),n=infinity) ); SetProperty('MCRaabe','value', Limit(n*(1-a(n+1)/a(n)),n=infinity) ); SetProperty( 'MCIntegral','value', Int(a(n),n=n0..infinity) ); SetProperty('Button2','enabled',false); SetProperty('MCResult','value', Sum(a(n),n=n0..infinity)); Do(%Plot0 = plot(0,n=0..20,color=black)); end use;

Convergence tests. From Wikipedia, the free encyclopedia. Convergence Tests for Infinite Series. From the "Online Mathematics Tutorials" at Harvey Mudd College. Choosing a Convergence Test for Infinite Series. David J. Manuel. List of Convergence Tests. From Wolfram MathWorld.

JSFH JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH