$sum_{n=1}^{infty} frac{(1/2) + (-1)^{n}}{n}$ converges or diverges?
How to check if the series $$sum_{n=1}^{infty} frac{(1/2) + (-1)^{n}}{n}$$ converges or diverges?
When $n$ is odd, series is $sum frac{-1}{2n}$
When $n$ is even, series is $sum frac{3}{2n}$
This series is similar to the series
$$sum frac{-1}{2(2n-1)} + frac{3}{2(2n)}$$
$$= sum frac{8n-6}{8n(2n-1)}$$
Which is clearly divergent.
So, the given series is divergent.
Is this method right?
Please, suggest if there is some easier way.
sequences-and-series
asked Dec 26 at 13:10
Mathsaddict
1778
add a comment |
How to check if the series $$sum_{n=1}^{infty} frac{(1/2) + (-1)^{n}}{n}$$ converges or diverges?
When $n$ is odd, series is $sum frac{-1}{2n}$
When $n$ is even, series is $sum frac{3}{2n}$
This series is similar to the series
$$sum frac{-1}{2(2n-1)} + frac{3}{2(2n)}$$
$$= sum frac{8n-6}{8n(2n-1)}$$
Which is clearly divergent.
So, the given series is divergent.
Is this method right?
Please, suggest if there is some easier way.
sequences-and-series
asked Dec 26 at 13:10
Mathsaddict
1778
Your method is not correct. You distinguish $n$ even and $n$ odd for the partial sum not for the general term.
– hamam_Abdallah
Dec 26 at 13:17
add a comment |
How to check if the series $$sum_{n=1}^{infty} frac{(1/2) + (-1)^{n}}{n}$$ converges or diverges?
When $n$ is odd, series is $sum frac{-1}{2n}$
When $n$ is even, series is $sum frac{3}{2n}$
This series is similar to the series
$$sum frac{-1}{2(2n-1)} + frac{3}{2(2n)}$$
$$= sum frac{8n-6}{8n(2n-1)}$$
Which is clearly divergent.
So, the given series is divergent.
Is this method right?
Please, suggest if there is some easier way.
sequences-and-series
asked Dec 26 at 13:10
Mathsaddict
1778
How to check if the series $$sum_{n=1}^{infty} frac{(1/2) + (-1)^{n}}{n}$$ converges or diverges?
When $n$ is odd, series is $sum frac{-1}{2n}$
When $n$ is even, series is $sum frac{3}{2n}$
This series is similar to the series
$$sum frac{-1}{2(2n-1)} + frac{3}{2(2n)}$$
$$= sum frac{8n-6}{8n(2n-1)}$$
Which is clearly divergent.
So, the given series is divergent.
Is this method right?
Please, suggest if there is some easier way.
sequences-and-series
sequences-and-series
asked Dec 26 at 13:10
Mathsaddict
1778
asked Dec 26 at 13:10
Mathsaddict
1778
asked Dec 26 at 13:10
Mathsaddict
1778
asked Dec 26 at 13:10
Mathsaddict
1778
asked Dec 26 at 13:10
Mathsaddict
1778
1778
Your method is not correct. You distinguish $n$ even and $n$ odd for the partial sum not for the general term.
– hamam_Abdallah
Dec 26 at 13:17
add a comment |
Your method is not correct. You distinguish $n$ even and $n$ odd for the partial sum not for the general term.
– hamam_Abdallah
Dec 26 at 13:17
Your method is not correct. You distinguish $n$ even and $n$ odd for the partial sum not for the general term.
– hamam_Abdallah
Dec 26 at 13:17
Your method is not correct. You distinguish $n$ even and $n$ odd for the partial sum not for the general term.
– hamam_Abdallah
Dec 26 at 13:17
add a comment |
2 Answers
2
active
oldest
votes
hint
$$sum frac{(-1)^n}{n}$$
is convergent by alternate criteria.
$$frac 12sum frac 1n$$
is a divergent Riemann series.
The sum of a convergent and a divergent series is a Divergent one.
answered Dec 26 at 13:14
hamam_Abdallah
38k21634
The sum should diverge. Right?
– Mathsaddict
Dec 26 at 13:24
@Mathsaddict Yes of course. but Div +Div could converge.
– hamam_Abdallah
Dec 26 at 13:25
add a comment |
The idea is correct, but not correctly expressed. Asserting that the given series converges is equivalent to the assertion that the sequence$$left(sum_{n=1}^Nfrac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges. If it does, then the sequence$$left(sum_{n=1}^{2N}frac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges too. But it follows from your computations that it doesn't.
answered Dec 26 at 13:16
José Carlos Santos
150k22120221
From my computations, sequence of partiat sum diverges, and so does the series.
– Mathsaddict
Dec 26 at 13:29
You write as if that's different from what I wrote.
– José Carlos Santos
Dec 26 at 13:41
I thought you are pointing towards some mistake in my calculations, since you wrote 'but' in the last line. Does your answer support my method to check if the series converges?
– Mathsaddict
Dec 26 at 13:52
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
hint
$$sum frac{(-1)^n}{n}$$
is convergent by alternate criteria.
$$frac 12sum frac 1n$$
is a divergent Riemann series.
The sum of a convergent and a divergent series is a Divergent one.
answered Dec 26 at 13:14
hamam_Abdallah
38k21634
The sum should diverge. Right?
– Mathsaddict
Dec 26 at 13:24
@Mathsaddict Yes of course. but Div +Div could converge.
– hamam_Abdallah
Dec 26 at 13:25
add a comment |
hint
$$sum frac{(-1)^n}{n}$$
is convergent by alternate criteria.
$$frac 12sum frac 1n$$
is a divergent Riemann series.
The sum of a convergent and a divergent series is a Divergent one.
answered Dec 26 at 13:14
hamam_Abdallah
38k21634
The sum should diverge. Right?
– Mathsaddict
Dec 26 at 13:24
@Mathsaddict Yes of course. but Div +Div could converge.
– hamam_Abdallah
Dec 26 at 13:25
add a comment |
hint
$$sum frac{(-1)^n}{n}$$
is convergent by alternate criteria.
$$frac 12sum frac 1n$$
is a divergent Riemann series.
The sum of a convergent and a divergent series is a Divergent one.
answered Dec 26 at 13:14
hamam_Abdallah
38k21634
hint
$$sum frac{(-1)^n}{n}$$
is convergent by alternate criteria.
$$frac 12sum frac 1n$$
is a divergent Riemann series.
The sum of a convergent and a divergent series is a Divergent one.
answered Dec 26 at 13:14
hamam_Abdallah
38k21634
edited Dec 26 at 13:42
answered Dec 26 at 13:14
hamam_Abdallah
38k21634
answered Dec 26 at 13:14
hamam_Abdallah
38k21634
answered Dec 26 at 13:14
hamam_Abdallah
38k21634
38k21634
The sum should diverge. Right?
– Mathsaddict
Dec 26 at 13:24
@Mathsaddict Yes of course. but Div +Div could converge.
– hamam_Abdallah
Dec 26 at 13:25
add a comment |
The sum should diverge. Right?
– Mathsaddict
Dec 26 at 13:24
@Mathsaddict Yes of course. but Div +Div could converge.
– hamam_Abdallah
Dec 26 at 13:25
The sum should diverge. Right?
– Mathsaddict
Dec 26 at 13:24
The sum should diverge. Right?
– Mathsaddict
Dec 26 at 13:24
@Mathsaddict Yes of course. but Div +Div could converge.
– hamam_Abdallah
Dec 26 at 13:25
@Mathsaddict Yes of course. but Div +Div could converge.
– hamam_Abdallah
Dec 26 at 13:25
add a comment |
The idea is correct, but not correctly expressed. Asserting that the given series converges is equivalent to the assertion that the sequence$$left(sum_{n=1}^Nfrac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges. If it does, then the sequence$$left(sum_{n=1}^{2N}frac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges too. But it follows from your computations that it doesn't.
answered Dec 26 at 13:16
José Carlos Santos
150k22120221
From my computations, sequence of partiat sum diverges, and so does the series.
– Mathsaddict
Dec 26 at 13:29
You write as if that's different from what I wrote.
– José Carlos Santos
Dec 26 at 13:41
I thought you are pointing towards some mistake in my calculations, since you wrote 'but' in the last line. Does your answer support my method to check if the series converges?
– Mathsaddict
Dec 26 at 13:52
add a comment |
The idea is correct, but not correctly expressed. Asserting that the given series converges is equivalent to the assertion that the sequence$$left(sum_{n=1}^Nfrac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges. If it does, then the sequence$$left(sum_{n=1}^{2N}frac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges too. But it follows from your computations that it doesn't.
answered Dec 26 at 13:16
José Carlos Santos
150k22120221
From my computations, sequence of partiat sum diverges, and so does the series.
– Mathsaddict
Dec 26 at 13:29
You write as if that's different from what I wrote.
– José Carlos Santos
Dec 26 at 13:41
I thought you are pointing towards some mistake in my calculations, since you wrote 'but' in the last line. Does your answer support my method to check if the series converges?
– Mathsaddict
Dec 26 at 13:52
add a comment |
The idea is correct, but not correctly expressed. Asserting that the given series converges is equivalent to the assertion that the sequence$$left(sum_{n=1}^Nfrac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges. If it does, then the sequence$$left(sum_{n=1}^{2N}frac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges too. But it follows from your computations that it doesn't.
answered Dec 26 at 13:16
José Carlos Santos
150k22120221
The idea is correct, but not correctly expressed. Asserting that the given series converges is equivalent to the assertion that the sequence$$left(sum_{n=1}^Nfrac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges. If it does, then the sequence$$left(sum_{n=1}^{2N}frac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges too. But it follows from your computations that it doesn't.
answered Dec 26 at 13:16
José Carlos Santos
150k22120221
answered Dec 26 at 13:16
José Carlos Santos
150k22120221
answered Dec 26 at 13:16
José Carlos Santos
150k22120221
answered Dec 26 at 13:16
José Carlos Santos
150k22120221
150k22120221
From my computations, sequence of partiat sum diverges, and so does the series.
– Mathsaddict
Dec 26 at 13:29
You write as if that's different from what I wrote.
– José Carlos Santos
Dec 26 at 13:41
I thought you are pointing towards some mistake in my calculations, since you wrote 'but' in the last line. Does your answer support my method to check if the series converges?
– Mathsaddict
Dec 26 at 13:52
add a comment |
From my computations, sequence of partiat sum diverges, and so does the series.
– Mathsaddict
Dec 26 at 13:29
You write as if that's different from what I wrote.
– José Carlos Santos
Dec 26 at 13:41
I thought you are pointing towards some mistake in my calculations, since you wrote 'but' in the last line. Does your answer support my method to check if the series converges?
– Mathsaddict
Dec 26 at 13:52
From my computations, sequence of partiat sum diverges, and so does the series.
– Mathsaddict
Dec 26 at 13:29
From my computations, sequence of partiat sum diverges, and so does the series.
– Mathsaddict
Dec 26 at 13:29
You write as if that's different from what I wrote.
– José Carlos Santos
Dec 26 at 13:41
You write as if that's different from what I wrote.
– José Carlos Santos
Dec 26 at 13:41
I thought you are pointing towards some mistake in my calculations, since you wrote 'but' in the last line. Does your answer support my method to check if the series converges?
– Mathsaddict
Dec 26 at 13:52
I thought you are pointing towards some mistake in my calculations, since you wrote 'but' in the last line. Does your answer support my method to check if the series converges?
– Mathsaddict
Dec 26 at 13:52
add a comment |
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Your method is not correct. You distinguish $n$ even and $n$ odd for the partial sum not for the general term.
– hamam_Abdallah
Dec 26 at 13:17