Lab 12

MAT 229, Spring 2021

Today’s lab is primarily about the ratio and the root tests for the convergence of a series. 

Your job: to explore these examples of uses (or attempted uses) of these tests.

Review

Absolute convergence test

If (Underoverscript[Σ, k, ∞])|(Subscript[b, k])| converges then  (Underoverscript[Σ, k, ∞])(Subscript[b, k]) converges.

Root and Ratio tests:

Given any series (Underoverscript[Σ, k, ∞])(Subscript[b, k]): evaluate the limit of either/or

the ratio of successive terms (ratio test), ignoring any signs:

L = (lim_(k->∞))|((Subscript[b, k+1])/(Subscript[b, k]))|

the (k^th) root, ignoring any signs (root test), of the (k^th) term:

L = (lim_(k->∞))((|(Subscript[b, k])|)^(1/k)) .

Then

If L<1, then (Underoverscript[Σ, k, ∞])(Subscript[b, k]) converges.

If  L>1, then (Underoverscript[Σ, k, ∞])(Subscript[b, k]) diverges.

If L =1, then you must use another convergence test. The series doesn’t compare to a geometric series.

Exercises to submit

Let’s start easy: suppose that you’re asked to consider the convergence of a geometric series:   (Underoverscript[Σ, (k=0), ∞])a (r^k).

a. Try the root test

b. Try the ratio test

c. State your conclusion. (What about the cases “on the boundary”?)

d. Was one test easier than the other?

Still easy, but maybe for the wrong reason: suppose that you’re asked to consider the convergence of the  series   (Underoverscript[Σ, (k=1), ∞])((((-1))^(k+1))/k).  We know that this is conditionally convergent (by the AST), but not absolutely convergent (since the harmonic series diverges). What do the root and ratio tests tell us?

a. Try the root test (you need to actually calculate the limit).

b. Try the ratio test (you need to actually calculate the limit).

c. What can you conclude from these tests?

d. Was one test easier than the other?

Still easy, but again for the wrong reason: suppose that you’re asked to consider the convergence of the  series   (Underoverscript[Σ, (k=1), ∞])(1/(k^2)).  We know that this is absolutely convergent (as a positive p-series). What do the root and ratio tests tell us?

a. Try the root test (you need to actually calculate the limit).

b. Try the ratio test (you need to actually calculate the limit).

c. What can you conclude from these tests?

d. Was one test easier than the other?

As we will soon see there are series representation for many well-known functions. However, if the series do not converge, the series cannot represent the function.

a. (e^x)=(Underoverscript[Σ, (k=0), ∞])((x^k)/(k!)), for any x where the series converges.

Determine if this series converges or not if x=-1. If it does use a partial sum, along with an appropriate error estimate, to approximate (e^(-1)) with error less than 0.0001.

Determine if this series converges or not if x=-2. If it does use a partial sum, along with an appropriate error estimate, to approximate (e^(-2)) with error less than 0.0001.

b. ln(x)=(((Underoverscript[Σ, (k=1), ∞])(-1))^(k+1))((((x-1))^k)/k), for any x where the series converges.

Determine if this series converges or not if x=0.5. If it does, use a partial sum, chosen using an appropriate error estimate, to approximate ln(0.5) with error less than 0.0001.

Determine if this series converges or not if x=2. If it does, use a partial sum, chosen using an appropriate error estimate, to approximate ln(2) with error less than 0.0001.