Absolute and Conditional Convergence Tests for Series | Homework Equations

[whatlifeforme]

Homework Statement

Absolute, Conditional, - convergence, or Divergence.

Homework Equations

\displaystyle \sum^{∞}_{n=1} (-1)^n e^{-n}

The Attempt at a Solution

1. Alternating Series Test
2. Ratio Test for ABsolute Convergence

1. \displaystyle (-1)^n (1/e)^n
an > 0 for n=1,2,3,4 - YES
decreasing - YES
limit (n->inf) = 0 - YES

Converges.

2.\displaystyle (1/e)^{n+1} * (e^n) = 1/e
limit (n->inf) = 1/e

1/e < 1

Thus, Absolute Convergence.

 

[Mark44]

Homework Statement

Absolute, Conditional, - convergence, or Divergence.

Homework Equations

\displaystyle \sum^{∞}_{n=1} (-1)^n e^{-n}

The Attempt at a Solution

1. Alternating Series Test
2. Ratio Test for ABsolute Convergence

1. \displaystyle (-1)^n (1/e)^n
an > 0 for n=1,2,3,4 - YES
decreasing - YES
limit (n->inf) = 0 - YES

Converges.

2.\displaystyle (1/e)^{n+1} * (e^n) = 1/e
limit (n->inf) = 1/e

1/e < 1

Thus, Absolute Convergence.

Yes, the series converges, and converges absolutely. Your work is a little sketchy in parts, so if your instructor is picky, you might lose points. For example, in the alt. series test you need to show that a_n > 0 for all n ≥ 1, not just n = 1, 2, 3, and 4. And you don't show any work that justifies your saying that the sequence a_n is decreasing or that the limit of the sequence is zero.

 

[whatlifeforme]

i did not show the work here, but i did show the work for decreasing by setting f(x) = series and taking derivative.
for the limit i have lim(x->inf) (1/e)^n = 0 (not sure how to show that this is zero)?

also, how do i show that an > 0 for all n >= 1 ?

 

[Mark44]

i did not show the work here, but i did show the work for decreasing by setting f(x) = series and taking derivative.
for the limit i have lim(x->inf) (1/e)^n = 0 (not sure how to show that this is zero)?

The limit as you wrote it doesn't make much sense, since you have x increasing, but the expression you're taking the limit of doesn't involve x.

One way to show that this limit is zero is to show that (1/e)ⁿ can be made arbitrarily close to zero. IOW, using the definition of the limit, which in this case involves N and $\epsilon$. I don't know if actual limit definitions have been presented in your class just yet.

also, how do i show that an > 0 for all n >= 1 ?

Here, a_n = 1/eⁿ. The only way for a rational expression to be equal to zero is for its numerator to equal zero. Obviously that can't happen here. The denominators in this sequence, e¹, e², ..., eⁿ, ... are an increasing sequence of positive numbers, so it must be true that their reciprocals are decreasing and positive. An induction proof, which would be easy, would be convincing here.

 

[whatlifeforme]

everytime we take a limit, i don't think the instructor wants us to use the definition of the proof. also, it's been a bit since i took a number theory class (discrete mathematics) which involved induction proofs.i stated what you did above about the denominators and their reciprocals, but i have no other proof than that. also, I'm not sure what i would say for my other problems.