- #1
Benny
- 584
- 0
[tex]
\sum\limits_{n = 1}^\infty {\frac{{\sin \left( {\frac{{n\pi }}{2}} \right)}}{{11 + 8n}}}
[/tex]
I know that the numerator oscillates between -1 and 1 but there are some values of n for which the sine term also takes on the value of zero. So I can't find an explicit form for the numerator which means I can't use the alternating series test. I can't think of any other tests to use since the expression inside the summation takes on negative values 'regularly.'
[tex]
\sum\limits_{n = 1}^\infty {\left( { - 1} \right)^n \frac{1}{{n^{1 + \frac{1}{n}} }}}
[/tex]
Hmm...this one is a bit trick so basically I just hoped that the alternating series test would yield something simple.
[tex]
n^{1 + \frac{1}{n}} \ge n^{1 + \frac{1}{{n + 1}}} \Leftrightarrow \frac{1}{{n^{1 + \frac{1}{n}} }} \le \frac{1}{{n^{1 + \frac{1}{{n + 1}}} }}
[/tex]
[tex]
a_n \le a_{n + 1}
[/tex]
The terms are not decreasing so the series diverges? My caculator suggests otherwise. Again, I'm not sure about this one.
Can someone help me out with these two series?
\sum\limits_{n = 1}^\infty {\frac{{\sin \left( {\frac{{n\pi }}{2}} \right)}}{{11 + 8n}}}
[/tex]
I know that the numerator oscillates between -1 and 1 but there are some values of n for which the sine term also takes on the value of zero. So I can't find an explicit form for the numerator which means I can't use the alternating series test. I can't think of any other tests to use since the expression inside the summation takes on negative values 'regularly.'
[tex]
\sum\limits_{n = 1}^\infty {\left( { - 1} \right)^n \frac{1}{{n^{1 + \frac{1}{n}} }}}
[/tex]
Hmm...this one is a bit trick so basically I just hoped that the alternating series test would yield something simple.
[tex]
n^{1 + \frac{1}{n}} \ge n^{1 + \frac{1}{{n + 1}}} \Leftrightarrow \frac{1}{{n^{1 + \frac{1}{n}} }} \le \frac{1}{{n^{1 + \frac{1}{{n + 1}}} }}
[/tex]
[tex]
a_n \le a_{n + 1}
[/tex]
The terms are not decreasing so the series diverges? My caculator suggests otherwise. Again, I'm not sure about this one.
Can someone help me out with these two series?