Section 11.5: Alternating Series

Review

Series

The infinite sum (Underoverscript[Σ, (k=1), ∞])(Subscript[a, k])=(Subscript[a, 1])+(Subscript[a, 2])+(Subscript[a, 3])+… converges or diverges if its partial sums converge or diverge.

Question

If the series (Underoverscript[Σ, (k=1), ∞])(Subscript[a, k]) converges, what must be true about the sequence (Subscript[a, k]), k=1,2,…?

Question

What particular series do we use for comparisons?

Question

What tests for convergence and diverges have we discussed?

Alternating series

So far we have looked mostly at series where we sum nonnegative terms. We want to explore what happens if some of the terms are negative.

Alternating series test

The alternating series (((Underoverscript[Σ, (k=1), ∞])(-1))^(k+1))(Subscript[a, k]) converges if

(Subscript[a, k])≥0 for k=1,2,3,…

The sequence {(Subscript[a, k])} is a decreasing sequence.

(lim_(k->∞))(Subscript[a, k])=0

Questions

Does the p-series (Underoverscript[Σ, (k=1), ∞])(1/k) converge or diverge? Why or why not?

Does the alternating p-series (((Underoverscript[Σ, (k=1), ∞])(-1))^(k+1))(1/k) converge or diverge? Why or why not?

Questions

What are the first four partial sums of the p-series (Underoverscript[Σ, (k=1), ∞])(1/k)?

p0=ListPlot[Table[{k,1/k},{k,1,10}],PlotLabels->"terms"];
p1=ListPlot[Table[{n,Sum[1/k,{k,1,n}]},{n,1,10}],PlotLabels->"partial sums"];
Show[p0,p1,PlotRange->All]

What are the first four partial sums the alternating p-series (((Underoverscript[Σ, (k=1), ∞])(-1))^k)(1/k)?

exact=Sum[(-1)^k/k,{k,1,Infinity}]
p0=ListPlot[Table[{k,1/k},{k,1,10}],PlotLabels->"un-signed terms"];
p1=ListPlot[Table[{n,Sum[(-1)^k/k,{k,1,n}]},{n,1,10}],PlotLabels->"partial sums"];
p2=ListLinePlot[{{0,exact},{10,exact}}];
Show[p0,p1,p2,PlotRange->All]

Questions

Does the alternating series (((Underoverscript[Σ, (k=1), ∞])(-1))^(k+1))(1/(k!)) converge or diverge? Why or why not?

Does the alternating series (((Underoverscript[Σ, (k=1), ∞])(-1))^(k+1))((k-1)/(k^2)) converge or diverge? Why or why not?

Notes

If the three conditions are met

the series is truly alternating,

the individual terms without the sign are decreasing, and

the limit of the individual terms go to zero,

then the alternating series converges.

If the last condition is not met, (lim_(n->∞))(Subscript[a, n])=0, then the series diverges by the divergence test.

If either of the first two conditions are not met, the alternating series test has no conclusions.

Questions

Does the series (((Underoverscript[Σ, (k=1), ∞])(-1))^(k+1))((k+1)/k) converge or diverge? Why or why not?

Does the series 1+(1/2)-(1/3)-(1/4)+(1/5)+(1/6)-(1/7)-(1/8)+… converge or diverge? Why or why not?

Does the series 1-(1/3)+(1/2)-(1/5)+(1/4)-(1/7)+(1/6)-(1/9)+… converge or diverge? Why or why not?

(* if you combine all terms in pairs after the first two, then *)
Plot[(2k^2+2k+1)/(k^2(k+1)^2),{k,1,10}]

Alternating series error estimate

Approximate series with partial sums. In the case of converging alternating series:
	(((Underoverscript[Σ, (k=1), ∞])(-1))^(k+1))(Subscript[a, k])≈(((Underoverscript[Σ, (k=1), n])(-1))^(k+1))(Subscript[a, k])=(Subscript[a, 1])-(Subscript[a, 2])+(Subscript[a, 3])-(Subscript[a, 4])+…+(((-1))^(n+1))(Subscript[a, n])

Questions

What is the error in approximating the alternating series (((Underoverscript[Σ, (k=1), ∞])(-1))^(k+1))(Subscript[a, k]) with its (n^th) partial sum?

What is an estimate on this error?

Alternating test error estimate

If the series (((Underoverscript[Σ, (k=1), ∞])(-1))^(k+1))(Subscript[a, k]) meets the requirements of the alternating series test so that it converges, then the error in approximating it with its (n^th) partial sum (((Underoverscript[Σ, (k=1), n])(-1))^(k+1))(Subscript[a, k]) is given very simply: 
						error≤(Subscript[a, n+1])

Questions

What is the error in approximating (((Underoverscript[Σ, (k=1), ∞])(-1))^(k+1))(1/((k^2)+3k+5)) with (((Underoverscript[Σ, (k=1), 10])(-1))^(k+1))(1/((k^2)+3k+5))? (check to make sure that the terms are decreasing first!)

Which partial sums approximate (((Underoverscript[Σ, (k=1), ∞])(-1))^(k+1))(1/k) with error less than 0.001?

Approximate (((Underoverscript[Σ, (k=1), ∞])(-1))^(k+1))(1/(k^3)) with error less than 0.001.

exact=Sum[(-1)^(k+1)/k^3,{k,1,Infinity}]
approx=Sum[(-1)^(k+1)/k^3,{k,1,10}]
p0=ListPlot[Table[{k,1/k^3},{k,1,10}],PlotLabels->"terms"];
p1=ListPlot[Table[{n,Sum[(-1)^(k+1)/k^3,{k,1,n}]},{n,1,10}],PlotLabels->"partial sums"];
p2=ListLinePlot[{{0,exact},{10,exact}}];
Show[p0,p1,p2,PlotRange->All]
error=N[Abs[approx-exact]]
(* Actually, we can get results "good enough" using only 8 in the partial sum: *)
approx=Sum[(-1)^(k+1)/k^3,{k,1,8}]
error=N[Abs[approx-exact]]