infinite series 1

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mario99

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Determine whether the series converges or diverges. If it converges, find the sum of the series.

\(\displaystyle \sum_{k=0}^{\infty} 3\left(\frac{1}{5}\right)^k\)

Any help would be appreciated!
 
mario99 said:
Determine whether the series converges or diverges. If it converges, find the sum of the series.

\(\displaystyle \sum_{k=0}^{\infty} 3\left(\frac{1}{5}\right)^k\)

Any help would be appreciated!
Click to expand...
Do you know how to derive the sum of infinite geometric series?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:


Please share your work/thoughts about this problem.
 
mario99 said:
Determine whether the series converges or diverges. If it converges, find the sum of the series.

\(\displaystyle \sum_{k=0}^{\infty} 3\left(\frac{1}{5}\right)^k\)

Any help would be appreciated!
Click to expand...
Please tell us what you have learned about geometric series, and show whatever work you can attempt. We want to help you work this out using what you know.

If you know nothing about it, start with a lesson on the topic, such as this.
 
Thank you very much Subhotosh Khan and Dr.Peterson for helping me.

I am guessing that this series diverges because it has no \(\displaystyle k\) inside it regardless of it is being geometric series or not.
 
mario99 said:
Thank you very much Subhotosh Khan and Dr.Peterson for helping me.

I am guessing that this series diverges because it has no \(\displaystyle k\) inside it regardless of it is being geometric series or not.
Click to expand...
Why do you think that? Of course there's a k there.

Please answer our questions.
 
mario99 said:
Determine whether the series converges or diverges. If it converges, find the sum of the series.

\(\displaystyle \sum_{k=0}^{\infty} 3\left(\frac{1}{5}\right)^k\)

Any help would be appreciated!
Click to expand...

You can rewrite this as:

\(\displaystyle 3\sum_{k=0}^{\infty} \left(\frac{1}{5}\right)^k\)

mario99 said:
I am guessing that this series diverges ...
Click to expand...

You could write out a partial sum from the way I rewrote it and make an educated guess as to whether it converges or diverges,
along with working the lessons presented in the link of post # 3.

For example:

\(\displaystyle 3[(1/5)^0 + (1/5)^1 + (1/5)^2 + (1/5)^3 + (1/5)^4 + (1/5)^5] \ = \ \)

\(\displaystyle 3[1 + 0.2 + 0.04 + 0.008 + 0.0016 + 0.00032] \ = \ \)

\(\displaystyle 3[1.24992] \ = \ \)

\(\displaystyle 3.74976\)
 
Thank you very much pka and lookagain for helping me.

Dr.Peterson said:
Why do you think that? Of course there's a k there.

Please answer our questions.
Click to expand...
My approach is that \(\displaystyle k\) will vanish after applying the root test.

\(\displaystyle \sum_{k=0}^{\infty}3\sqrt[k]{\left(\frac{1}{5}\right)^k} = \sum_{k=0}^{\infty}3\frac{1}{5} = 3\frac{1}{5} \times \infty = \infty\)


pka said:
Please look at this link.
Click to expand...
You are using a computer. The steps that were used to get this answer were not obvious. I am following human methods.


lookagain said:
You could write out a partial sum from the way I rewrote it and make an educated guess as to whether it converges or diverges,
Click to expand...
Although it seems a nice method, I can't make an educated guess. I need to be an expert to be able to think like that.
 
mario99 said:
My approach is that \(\displaystyle k\) will vanish after applying the root test.

\(\displaystyle \sum_{k=0}^{\infty}3\sqrt[k]{\left(\frac{1}{5}\right)^k} = \sum_{k=0}^{\infty}3\frac{1}{5} = 3\frac{1}{5} \times \infty = \infty\)
Click to expand...
But that's not what the root test says to do:

Calculus II - Root Test

In this section we will discuss using the Root Test to determine if an infinite series converges absolutely or diverges. The Root Test can be used on any series, but unfortunately will not always yield a conclusive answer as to whether a series will converge absolutely or diverge. A proof of...
tutorial.math.lamar.edu

It doesn't involve summing the roots, just taking the limit.

Try that again.

Thanks for showing work, so we have things to talk about. The more you say, the better we can help.
 
You're welcome Dr.Peterson.

Oh I forgot the limit part.

\(\displaystyle \lim_{k\to\infty}\sqrt[k]{\left(\frac{1}{5}\right)^k} = \frac{1}{5}\)

Now I am lost. What can this tell me?
 
mario99 said:
You're welcome Dr.Peterson.

Oh I forgot the limit part.

\(\displaystyle \lim_{k\to\infty}\sqrt[k]{\left(\frac{1}{5}\right)^k} = \frac{1}{5}\)

Now I am lost. What can this tell me?
Click to expand...
Read the link I gave you to see what this implies. It's right there at the top.
 
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