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Determine whether the series converges or diverges.

  • Thread starter Sabricd
  • Start date
  • #1
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Hello,

I have to determine whether the given series diverges or converges. [tex]\sum[/tex] cos(n*pi)/(n^(3/4)) where n= 1 and goes to infinity.

I tried a couple numbers for n and got:

-1 + 1/(2)^(3/4) - 1/(3)^(3/4)

Hence I came up with the series: [tex]\sum[/tex]((-1)^n)/(n)^(3/4) where n=1 and goes to infinity.

I guess my main question is that now that I have that new representation of the series, why can't I just take the absolute value of the series and say that it is a p series with p< 1 and therefore diverges.

My book took the limit of the series and got 0 and said it converged. Why can't you use the p-series test???


Thank you,
 

Answers and Replies

  • #3
27
0
So when you have alternating series you cannot use the p-series test???
 
  • #4
22,097
3,282
But the p-series is for series of the form

[tex]\sum_{n=1}^\infty{\frac{1}{n^p}}[/tex]

You have a factor [tex](-1)^n[/tex] extra.
I guess that you could say that you just took the absolute value of each term. But it is NOT TRUE to say that

[tex]\sum{x_n}~\text{diverges}~\text{if}~\sum{|x_n|}~\text{diverges}[/tex]

if you replace diverges with converges, then it is true however...
 

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