Find all values of c such that sum[k=0,infty][2/(ck+1)] will converge

[rcoggin]

The question states (verbatim): Determine all values of c such that the series converges. Then gives the below series.

. . . . .$\displaystyle \displaystyle \sum_{k\, =\, 0}^{\infty}\,$$\displaystyle \dfrac{2}{ck\, +\, 1}$

This one has been plaguing me for going on three days. It is the last problem in my current homework assignment, and I cannot figure out how to come up with any value of c that makes this series converge. I have gone through comparison tests, the integral test, ratio, etc., but I keep basically arriving at either infinity or DNE. My textbook is zero help as it does not even have one example even remotely similar to this problem.

Can anyone point me in the right direction? Or is it in fact, unable to converge at any value of c?

Thanks in advance!

Robert

 

[stapel]

The question states (verbatim): Determine all values of c such that the series converges. Then gives the below series.

. . . . .$\displaystyle \displaystyle \sum_{k\, =\, 0}^{\infty}\,$$\displaystyle \dfrac{2}{ck\, +\, 1}$

You know that the harmonic series:

. . . . .$\displaystyle \displaystyle \sum_{k\, =\, 1}^{\infty}\,$$\displaystyle \dfrac{1}{k}$

...diverges. This series may also be stated as:

. . . . .$\displaystyle \displaystyle \sum_{k\, =\, 0}^{\infty}\,$$\displaystyle \dfrac{1}{k\, +\, 1}$

...by adjusting the parameter. Since this diverges, then so also does:

. . . . .$\displaystyle \displaystyle \sum_{k\, =\, 0}^{\infty}\,$$\displaystyle 2\, \left(\dfrac{1}{k\, +\, 1}\right)\, =\,$$\displaystyle \displaystyle \sum_{k\, =\, 0}^{\infty}\,$$\displaystyle \dfrac{2}{k\, +\, 1}$

...since multiplying "goes to infinity" by two leaves the divergence intact. Now, letting:

. . . . .$\displaystyle \dfrac{2}{k\, +\, 1}\, =\, a_k$

...and:

. . . . .$\displaystyle \dfrac{2}{ck\, +\, 1}\, =\, b_k$

...can you think of any convergence tests which might be helpful?

 

[pka]

The question states (verbatim): Determine all values of c such that the series converges. Then gives the below series.
$\displaystyle \displaystyle \sum_{k\, =\, 0}^{\infty}\,$$\displaystyle \dfrac{2}{ck\, +\, 1}$
My textbook is zero help as it does not even have one example even remotely similar to this problem.

If $\displaystyle \large c=0$ we are adding infinitely many two's.

If $\displaystyle \large c\ne 0$ then the sum is $\displaystyle \displaystyle\large{\frac{1}{c}\sum\limits_{k = 0}^\infty {\dfrac{2}{{k + {c^{ - 1}}}}}}$ is that a harmonic series?

 

[rcoggin]

Well, set up as you have shown it

b_k would be > a_k for all values of c > 0 and the opposite if c < or equal to 0. Since we know 2/k+1 diverges, if b_k < a_k with c < or = 0, then a_k would also diverge by the comparison test. Is this a correct assumption?

Given the establishment of an a_k and b_k this points to either the comparison test or the limit comparison test. I tried both of those but could not get any kind of conclusive answer. Not sure if I did them wrong. I'll have to see how to show my work in here...

Thank you!

 

[rcoggin]

If $\displaystyle \large c=0$ we are adding infinitely many two's.

If $\displaystyle \large c\ne 0$ then the sum is $\displaystyle \displaystyle\large{\frac{1}{c}\sum\limits_{k = 0}^\infty {\dfrac{2}{{k + {c^{ - 1}}}}}}$ is that a harmonic series?

As c is a constant I would say yes, this is a harmonic series as "2c" could be pulled out front. Then it would be divergent. I've already been down this road, and am trying to prove that it can converge with some value of c.

 

[pka]

As c is a constant I would say yes, this is a harmonic series as "2c" could be pulled out front. Then it would be divergent. I've already been down this road, and am trying to prove that it can converge with some value of c.

NO! there is no other road to go down.

I have shown you that it diverges for every real number c.

 

[rcoggin]

Thanks. I'm new to this stuff and just now learning it otherwise I wouldn't be here asking for help. The question that was given to me seems misleading and suggests a possible solution, and I am trying to make sure that there hasn't been something I've been missing in my attempts, or am not yet understanding. I don't know how to post the math type directly in the text of my post, so here's a photo of my work. I also performed the integral test and a ratio test previous to this. Thank you for your time.

 

[stapel]

Thank you for showing your work so nicely. I'd used the Limit Comparison Test to prove the divergence of what they'd given you. Nice job!