Calculating $sumlimits_{n=1}^inftyfrac{{1}}{n+3^n} $
I was able to prove this sum
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}$$
is convergent through the comparison test but I don't get how to find its sum.
calculus sequences-and-series convergence summation
|
show 11 more comments
I was able to prove this sum
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}$$
is convergent through the comparison test but I don't get how to find its sum.
calculus sequences-and-series convergence summation
2
@FordDavis Ask your teacher about the question again. It's quite unlikely they expect for you to find the sum. Or maybe the question was $frac{1}{n3^n}$, which has a much nicer answer.
– Toby Mak
Dec 2 at 0:21
2
What about this one: Is there any non-zero function $f(n) $ such that $sum_{n=1}^infty frac{1} {f(n) +3^n}$ has an okay closed form?
– Zacky
Dec 2 at 0:22
1
@Zacky There are some trivial choices like $f(n) = 2^n - 3^n$.
– Daniel
Dec 2 at 0:33
1
There's something odd about this assignment. Part b) can be summed in closed form, but it takes techniques way beyond calc 2. Are you perhaps supposed to approximate the sums. I definitely second @TobyMak's advice: ask your teacher about it.
– saulspatz
Dec 2 at 0:33
3
As noted by Carl, this question already exists ... math.stackexchange.com/q/1672121/442 but even that has no answer in nearly 3 years. No answer = cannot mark this as a duplicate.
– GEdgar
Dec 2 at 1:14
|
show 11 more comments
I was able to prove this sum
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}$$
is convergent through the comparison test but I don't get how to find its sum.
calculus sequences-and-series convergence summation
I was able to prove this sum
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}$$
is convergent through the comparison test but I don't get how to find its sum.
calculus sequences-and-series convergence summation
calculus sequences-and-series convergence summation
edited Dec 2 at 0:51
David G. Stork
9,73921232
9,73921232
asked Dec 2 at 0:05
Ford Davis
493
493
2
@FordDavis Ask your teacher about the question again. It's quite unlikely they expect for you to find the sum. Or maybe the question was $frac{1}{n3^n}$, which has a much nicer answer.
– Toby Mak
Dec 2 at 0:21
2
What about this one: Is there any non-zero function $f(n) $ such that $sum_{n=1}^infty frac{1} {f(n) +3^n}$ has an okay closed form?
– Zacky
Dec 2 at 0:22
1
@Zacky There are some trivial choices like $f(n) = 2^n - 3^n$.
– Daniel
Dec 2 at 0:33
1
There's something odd about this assignment. Part b) can be summed in closed form, but it takes techniques way beyond calc 2. Are you perhaps supposed to approximate the sums. I definitely second @TobyMak's advice: ask your teacher about it.
– saulspatz
Dec 2 at 0:33
3
As noted by Carl, this question already exists ... math.stackexchange.com/q/1672121/442 but even that has no answer in nearly 3 years. No answer = cannot mark this as a duplicate.
– GEdgar
Dec 2 at 1:14
|
show 11 more comments
2
@FordDavis Ask your teacher about the question again. It's quite unlikely they expect for you to find the sum. Or maybe the question was $frac{1}{n3^n}$, which has a much nicer answer.
– Toby Mak
Dec 2 at 0:21
2
What about this one: Is there any non-zero function $f(n) $ such that $sum_{n=1}^infty frac{1} {f(n) +3^n}$ has an okay closed form?
– Zacky
Dec 2 at 0:22
1
@Zacky There are some trivial choices like $f(n) = 2^n - 3^n$.
– Daniel
Dec 2 at 0:33
1
There's something odd about this assignment. Part b) can be summed in closed form, but it takes techniques way beyond calc 2. Are you perhaps supposed to approximate the sums. I definitely second @TobyMak's advice: ask your teacher about it.
– saulspatz
Dec 2 at 0:33
3
As noted by Carl, this question already exists ... math.stackexchange.com/q/1672121/442 but even that has no answer in nearly 3 years. No answer = cannot mark this as a duplicate.
– GEdgar
Dec 2 at 1:14
2
2
@FordDavis Ask your teacher about the question again. It's quite unlikely they expect for you to find the sum. Or maybe the question was $frac{1}{n3^n}$, which has a much nicer answer.
– Toby Mak
Dec 2 at 0:21
@FordDavis Ask your teacher about the question again. It's quite unlikely they expect for you to find the sum. Or maybe the question was $frac{1}{n3^n}$, which has a much nicer answer.
– Toby Mak
Dec 2 at 0:21
2
2
What about this one: Is there any non-zero function $f(n) $ such that $sum_{n=1}^infty frac{1} {f(n) +3^n}$ has an okay closed form?
– Zacky
Dec 2 at 0:22
What about this one: Is there any non-zero function $f(n) $ such that $sum_{n=1}^infty frac{1} {f(n) +3^n}$ has an okay closed form?
– Zacky
Dec 2 at 0:22
1
1
@Zacky There are some trivial choices like $f(n) = 2^n - 3^n$.
– Daniel
Dec 2 at 0:33
@Zacky There are some trivial choices like $f(n) = 2^n - 3^n$.
– Daniel
Dec 2 at 0:33
1
1
There's something odd about this assignment. Part b) can be summed in closed form, but it takes techniques way beyond calc 2. Are you perhaps supposed to approximate the sums. I definitely second @TobyMak's advice: ask your teacher about it.
– saulspatz
Dec 2 at 0:33
There's something odd about this assignment. Part b) can be summed in closed form, but it takes techniques way beyond calc 2. Are you perhaps supposed to approximate the sums. I definitely second @TobyMak's advice: ask your teacher about it.
– saulspatz
Dec 2 at 0:33
3
3
As noted by Carl, this question already exists ... math.stackexchange.com/q/1672121/442 but even that has no answer in nearly 3 years. No answer = cannot mark this as a duplicate.
– GEdgar
Dec 2 at 1:14
As noted by Carl, this question already exists ... math.stackexchange.com/q/1672121/442 but even that has no answer in nearly 3 years. No answer = cannot mark this as a duplicate.
– GEdgar
Dec 2 at 1:14
|
show 11 more comments
1 Answer
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oldest
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What we can obtain, also by hand, is a reasonable estimation for example by
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}approx 0.392<sum_{n=1}^3frac{{1}}{n+3^n}+sum_{n=4}^inftyfrac{{1}}{3^n}=$$$$=frac14+frac1{11}+frac1{30}+frac32-1-frac13-frac1{9}-frac1{27}approx 0.393$$
I dont remember us going over approximating the sum, but this might be what she wanted. Thanks for your help.
– Ford Davis
Dec 2 at 2:40
@FordDavis Yes indeed in that case seems convenient use the fact that $3^n$ is larger than $n$ also for $n$ small to obtain a god estimation for the series.
– gimusi
Dec 2 at 9:13
add a comment |
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1 Answer
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What we can obtain, also by hand, is a reasonable estimation for example by
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}approx 0.392<sum_{n=1}^3frac{{1}}{n+3^n}+sum_{n=4}^inftyfrac{{1}}{3^n}=$$$$=frac14+frac1{11}+frac1{30}+frac32-1-frac13-frac1{9}-frac1{27}approx 0.393$$
I dont remember us going over approximating the sum, but this might be what she wanted. Thanks for your help.
– Ford Davis
Dec 2 at 2:40
@FordDavis Yes indeed in that case seems convenient use the fact that $3^n$ is larger than $n$ also for $n$ small to obtain a god estimation for the series.
– gimusi
Dec 2 at 9:13
add a comment |
What we can obtain, also by hand, is a reasonable estimation for example by
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}approx 0.392<sum_{n=1}^3frac{{1}}{n+3^n}+sum_{n=4}^inftyfrac{{1}}{3^n}=$$$$=frac14+frac1{11}+frac1{30}+frac32-1-frac13-frac1{9}-frac1{27}approx 0.393$$
I dont remember us going over approximating the sum, but this might be what she wanted. Thanks for your help.
– Ford Davis
Dec 2 at 2:40
@FordDavis Yes indeed in that case seems convenient use the fact that $3^n$ is larger than $n$ also for $n$ small to obtain a god estimation for the series.
– gimusi
Dec 2 at 9:13
add a comment |
What we can obtain, also by hand, is a reasonable estimation for example by
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}approx 0.392<sum_{n=1}^3frac{{1}}{n+3^n}+sum_{n=4}^inftyfrac{{1}}{3^n}=$$$$=frac14+frac1{11}+frac1{30}+frac32-1-frac13-frac1{9}-frac1{27}approx 0.393$$
What we can obtain, also by hand, is a reasonable estimation for example by
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}approx 0.392<sum_{n=1}^3frac{{1}}{n+3^n}+sum_{n=4}^inftyfrac{{1}}{3^n}=$$$$=frac14+frac1{11}+frac1{30}+frac32-1-frac13-frac1{9}-frac1{27}approx 0.393$$
edited Dec 2 at 9:14
answered Dec 2 at 2:24
gimusi
1
1
I dont remember us going over approximating the sum, but this might be what she wanted. Thanks for your help.
– Ford Davis
Dec 2 at 2:40
@FordDavis Yes indeed in that case seems convenient use the fact that $3^n$ is larger than $n$ also for $n$ small to obtain a god estimation for the series.
– gimusi
Dec 2 at 9:13
add a comment |
I dont remember us going over approximating the sum, but this might be what she wanted. Thanks for your help.
– Ford Davis
Dec 2 at 2:40
@FordDavis Yes indeed in that case seems convenient use the fact that $3^n$ is larger than $n$ also for $n$ small to obtain a god estimation for the series.
– gimusi
Dec 2 at 9:13
I dont remember us going over approximating the sum, but this might be what she wanted. Thanks for your help.
– Ford Davis
Dec 2 at 2:40
I dont remember us going over approximating the sum, but this might be what she wanted. Thanks for your help.
– Ford Davis
Dec 2 at 2:40
@FordDavis Yes indeed in that case seems convenient use the fact that $3^n$ is larger than $n$ also for $n$ small to obtain a god estimation for the series.
– gimusi
Dec 2 at 9:13
@FordDavis Yes indeed in that case seems convenient use the fact that $3^n$ is larger than $n$ also for $n$ small to obtain a god estimation for the series.
– gimusi
Dec 2 at 9:13
add a comment |
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2
@FordDavis Ask your teacher about the question again. It's quite unlikely they expect for you to find the sum. Or maybe the question was $frac{1}{n3^n}$, which has a much nicer answer.
– Toby Mak
Dec 2 at 0:21
2
What about this one: Is there any non-zero function $f(n) $ such that $sum_{n=1}^infty frac{1} {f(n) +3^n}$ has an okay closed form?
– Zacky
Dec 2 at 0:22
1
@Zacky There are some trivial choices like $f(n) = 2^n - 3^n$.
– Daniel
Dec 2 at 0:33
1
There's something odd about this assignment. Part b) can be summed in closed form, but it takes techniques way beyond calc 2. Are you perhaps supposed to approximate the sums. I definitely second @TobyMak's advice: ask your teacher about it.
– saulspatz
Dec 2 at 0:33
3
As noted by Carl, this question already exists ... math.stackexchange.com/q/1672121/442 but even that has no answer in nearly 3 years. No answer = cannot mark this as a duplicate.
– GEdgar
Dec 2 at 1:14