Absolute Convergence, and Root and Ratio Tests

MAT 229, Spring 2025

Supporting materials

If you wish to get a different perspective on the notes below, try either of the following textbook sections.

Strang’s Calculus
	Vol 2, Section 5.5: Alternating Series (Absolute convergence)
	Vol 2, Section 5.6: Ratio and root tests

Stewart’s Calculus
	Section 11.6: Absolute convergence and the ratio and root tests

Boelkins/Austin/Schlicker’s Active Calculus
	Section 8.3: Series of real numbers

Review

Question

The alternating series test guarantees (((Underoverscript[Σ, (k=1), ∞])(-1))^(k+1))(Subscript[a, k]) converges if what is true about (Subscript[a, k])? (Three things....)

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

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

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

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

Alternating series error estimate

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 easy estimate on this error?

Questions

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

Clear[n]
Solve[0.001==1/(n+1),n]
Sum[N[(-1)^k/k],{k,1,999}]
N[-Log[2]]

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

Clear[n]
Solve[0.001==1/(n+1)^3,n]
Sum[(-1)^k/k^3,{k,1,9}]
N[%]
Sum[(-1)^k/k^3,{k,1,Infinity}]
N[%]

Absolute convergence test

What if a series has some positive and some negative terms and neither the divergence test nor the alternating series test are applicable?

Absolute convergence test

If (Underoverscript[Σ, k, ∞])|(Subscript[b, k])| converges then  (Underoverscript[Σ, k, ∞])(Subscript[b, k]) converges, and we say that it converges absolutely. If (Underoverscript[Σ, k, ∞])|(Subscript[b, k])| diverges but  (Underoverscript[Σ, k, ∞])(Subscript[b, k]) converges, then  (Underoverscript[Σ, k, ∞])(Subscript[b, k]) converges, but it does so only conditionally.

A given series either converges absolutely, converges conditionally, or it diverges.

Question

We have discussed a series that meets the definition of conditional convergence. What is it?

Question

Determine whether the series converges absolutely, converges conditionally, or diverges.

(((Underoverscript[Σ, (k=1), ∞])(-1))^(k+1))(k/((k^2)+1))

(((Underoverscript[Σ, (k=0), ∞])(-1))^k)((e^k)/(k+1))

(((Underoverscript[Σ, (k=1), ∞])(-1))^(k-1))((3(k^2)-1)/(5(k^5)+(k^3)-2k))

Questions

Consider the series 1+(1/4)-(1/9)-(1/16)+(1/25)+(1/36)-(1/49)-(1/64)+…

Does the divergence test imply anything about this sequence?

Is the alternating series test applicable?

Is the absolute convergence test applicable?

Does this series converge or diverge?

Sum[1/k^2,{k,1,Infinity}]

Questions

Consider the series (Underoverscript[Σ, (k=1), ∞])((cos(k))/(k^3)).

Begin by applying the absolute convergence test to look at (Underoverscript[Σ, (k=1), ∞])((|cos(k)|)/(k^3)). What are the first 5 partial sums of this series?

Comparing this series to one you are familiar with, can you find a bound for the sequence of partial sums?

N[Sum[1/k^3,{k,1,Infinity}]]

Does (Underoverscript[Σ, (k=1), ∞])((|cos(k)|)/(k^3)) converge?

Does (Underoverscript[Σ, (k=1), ∞])((cos(k))/(k^3)) converge?

How can we estimate the error in approximating (Underoverscript[Σ, (k=1), ∞])((cos(k))/(k^3)) with (Underoverscript[Σ, (k=1), 10])((cos(k))/(k^3))?

Ratio and root tests

Two tests that will be the workhorses for our future analyses are based on comparisons to geometric series. They incorporate the absolute convergence test.

Questions

What is the form of a geometric series?

Under what circumstances does a geometric series converge or diverge?

Questions

Let (Subscript[c, k]) be the (k^th) term of a geometric series. What is ((Subscript[c, k+1])/(Subscript[c, k]))? What is (lim_(k->∞))((Subscript[c, k+1])/(Subscript[c, k]))? What about (lim_(k->∞))((Subscript[c, k+1])/(Subscript[c, k])) Implies the convergence of (Underoverscript[Σ, k, ∞])(Subscript[c, k])?

Let (Subscript[c, k]) be the (k^th) term of a geometric series. What is ((Subscript[c, k])^(1/k))? What is (lim_(k->∞))((Subscript[c, k])^(1/k))? What about (lim_(k->∞))((Subscript[c, k])^(1/k)) Implies the convergence of (Underoverscript[Σ, k, ∞])(Subscript[c, k])?

Ratio test

Given any series (Underoverscript[Σ, k, ∞])(Subscript[b, k]), evaluate the limit of the ratio of successive terms, ignoring any signs, (lim_(k->∞))|((Subscript[b, k+1])/(Subscript[b, k]))|.

If (lim_(k->∞))|((Subscript[b, k+1])/(Subscript[b, k]))|=L where L<1, then (Underoverscript[Σ, k, ∞])(Subscript[b, k]) converges.

If (lim_(k->∞))|((Subscript[b, k+1])/(Subscript[b, k]))|=L where L>1, then (Underoverscript[Σ, k, ∞])(Subscript[b, k]) diverges.

If (lim_(k->∞))|((Subscript[b, k+1])/(Subscript[b, k]))| =1, then you must use another convergence test. The series doesn’t compare to a geometric series.

Questions

Apply the ratio test to determine the convergence of the following series.

(Underoverscript[Σ, (k=1), ∞])(k/(4^k))

(Underoverscript[Σ, (k=1), ∞])((((-3))^k)/(k!))

Root test

Given any series (Underoverscript[Σ, k, ∞])(Subscript[b, k]), evaluate the limit of the (k^th) root, ignoring any signs, of the (k^th) term, (lim_(k->∞))((|(Subscript[b, k])|)^(1/k)).

If (lim_(k->∞))((|(Subscript[b, k])|)^(1/k))=L where L<1, then (Underoverscript[Σ, k, ∞])(Subscript[b, k]) converges.

If (lim_(k->∞))((|(Subscript[b, k])|)^(1/k))=L where L>1, then (Underoverscript[Σ, k, ∞])(Subscript[b, k]) diverges.

If (lim_(k->∞))((|(Subscript[b, k])|)^(1/k))=L where L=1, then you must use another convergence test. The series doesn’t compare to a geometric series.

Questions

Apply the root test to determine the convergence of the following series.

(Underoverscript[Σ, (k=1), ∞])(k/(4^k))

(((Underoverscript[Σ, (k=1), ∞])(-1))^(k+1))(((((k^2)+k +3))^k)/(100^k))