Sum of first n terms *DIFFICULT*

[sammmiewong]

Anyone can help with the following question? I believe this is a geometric series but I can't find r. Thanks a lot!

 

[Dr.Peterson]

No, it is not a geometric series. Why do you think that, if you know there is no "r" (that is, the ratio is not constant)?

What techniques have you been taught for summing series? What is the source of this problem?

 

[sammmiewong]

https://www.toppr.com/guides/maths/sequences-and-series/special-series/

But the answer is wrong here.

I was trying to write this series in a summation in terms of n but it wouldn't work either. The fraction cannot be simplified.

I thought this is a geometric because the numerator multiplies by x^2 and denominator multiplies by something.

 

[Dr.Peterson]

https://www.toppr.com/guides/maths/sequences-and-series/special-series/

But the answer is wrong here.

I was trying to write this series in a summation in terms of n but it wouldn't work either. The fraction cannot be simplified.

I thought this is a geometric because the numerator multiplies by x^2 and denominator multiplies by something.

I don't see this problem on that page; and the page is not only poorly organized, but full of wrong statements. Ignore it.

Writing a series as a summation, and evaluating the sum, are two entirely different things. Can you show me what you did?

A geometric sequence is one in which the ratio of successive terms is a constant, not just "something".

What have you actually learned about series, apart from looking at random, poorly written web pages?

 

[pka]

I was trying to write this series in a summation in terms of n but it wouldn't work either. The fraction cannot be simplified.

Here is the series: \(\displaystyle\sum\limits_{k = 0}^\infty {\frac{{{x^{2k}}}}{{(1 - {x^{2k + 1}})(1 - {x^{2k + 3}})}}} \)

 

[sammmiewong]

Here is the series: \(\displaystyle\sum\limits_{k = 0}^\infty {\frac{{{x^{2k}}}}{{(1 - {x^{2k + 1}})(1 - {x^{2k + 3}})}}} \)

Yes that’s what I got and cannot be further simplified.

So first of all I tried to do this question by using normal AP / GP method, looking for d / r but failed. Then I tried to rewrite the series into a summation and thinking perhaps this could be simplified and might lead me closer to the answer. But that failed too. So I’m stuck. Do you think this is a faulty question for AP / GP topic? Is there other method to do this? Many thanks.

 

[sammmiewong]

I don't see this problem on that page; and the page is not only poorly organized, but full of wrong statements. Ignore it.

Writing a series as a summation, and evaluating the sum, are two entirely different things. Can you show me what you did?

A geometric sequence is one in which the ratio of successive terms is a constant, not just "something".

What have you actually learned about series, apart from looking at random, poorly written web pages?

I just checked the site again. They have removed this question. Weird. So I guess this should not be under this AP / GP topic? Please shed some light on other method to solve this question. Thanks again.