Sum of a (finite) hyperharmonic series
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For some $k < infty$, and $p in (0,1)$, consider the following sum:
$S_k = sum_{n=1}^{k} frac{1}{n^p}$.
What is a closed form expression for $S_k$?
integration sequences-and-series power-series
asked Nov 16 at 6:08
math_phile
123
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up vote
0
down vote
favorite
For some $k < infty$, and $p in (0,1)$, consider the following sum:
$S_k = sum_{n=1}^{k} frac{1}{n^p}$.
What is a closed form expression for $S_k$?
integration sequences-and-series power-series
asked Nov 16 at 6:08
math_phile
123
It may worth have a look at Hurwitz's zeta function.
– Kemono Chen
Nov 16 at 6:12
@KemonoChen: Hurwitz's zeta function is also infinite, check generalized harmonic numbers.
– Tianlalu
Nov 16 at 6:17
@Tianlalu You can make a subtraction from Riemann's zeta function.
– Kemono Chen
Nov 16 at 6:20
Thanks guys. I found the answer below helpful: math.stackexchange.com/questions/451558/…
– math_phile
Nov 16 at 6:45
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
For some $k < infty$, and $p in (0,1)$, consider the following sum:
$S_k = sum_{n=1}^{k} frac{1}{n^p}$.
What is a closed form expression for $S_k$?
integration sequences-and-series power-series
asked Nov 16 at 6:08
math_phile
123
For some $k < infty$, and $p in (0,1)$, consider the following sum:
$S_k = sum_{n=1}^{k} frac{1}{n^p}$.
What is a closed form expression for $S_k$?
integration sequences-and-series power-series
integration sequences-and-series power-series
asked Nov 16 at 6:08
math_phile
123
asked Nov 16 at 6:08
math_phile
123
asked Nov 16 at 6:08
math_phile
123
asked Nov 16 at 6:08
math_phile
123
asked Nov 16 at 6:08
math_phile
123
123
It may worth have a look at Hurwitz's zeta function.
– Kemono Chen
Nov 16 at 6:12
@KemonoChen: Hurwitz's zeta function is also infinite, check generalized harmonic numbers.
– Tianlalu
Nov 16 at 6:17
@Tianlalu You can make a subtraction from Riemann's zeta function.
– Kemono Chen
Nov 16 at 6:20
Thanks guys. I found the answer below helpful: math.stackexchange.com/questions/451558/…
– math_phile
Nov 16 at 6:45
add a comment |
It may worth have a look at Hurwitz's zeta function.
– Kemono Chen
Nov 16 at 6:12
@KemonoChen: Hurwitz's zeta function is also infinite, check generalized harmonic numbers.
– Tianlalu
Nov 16 at 6:17
@Tianlalu You can make a subtraction from Riemann's zeta function.
– Kemono Chen
Nov 16 at 6:20
Thanks guys. I found the answer below helpful: math.stackexchange.com/questions/451558/…
– math_phile
Nov 16 at 6:45
It may worth have a look at Hurwitz's zeta function.
– Kemono Chen
Nov 16 at 6:12
It may worth have a look at Hurwitz's zeta function.
– Kemono Chen
Nov 16 at 6:12
@KemonoChen: Hurwitz's zeta function is also infinite, check generalized harmonic numbers.
– Tianlalu
Nov 16 at 6:17
@KemonoChen: Hurwitz's zeta function is also infinite, check generalized harmonic numbers.
– Tianlalu
Nov 16 at 6:17
@Tianlalu You can make a subtraction from Riemann's zeta function.
– Kemono Chen
Nov 16 at 6:20
@Tianlalu You can make a subtraction from Riemann's zeta function.
– Kemono Chen
Nov 16 at 6:20
Thanks guys. I found the answer below helpful: math.stackexchange.com/questions/451558/…
– math_phile
Nov 16 at 6:45
Thanks guys. I found the answer below helpful: math.stackexchange.com/questions/451558/…
– math_phile
Nov 16 at 6:45
add a comment |
1 Answer
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I am not aware of any closed form, but there are some good approximations. The easiest way would be to approximate this sum with an integral:
$$sum_{n=1}^kn^{-p}approx1+int_1^k x^{-p}dx=1+left.frac1{1-p}x^{1-p}rightrvert^k_0=1+frac{k^{1-p}}{1-p}.$$
answered Nov 16 at 7:33
YiFan
1,6891314
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
I am not aware of any closed form, but there are some good approximations. The easiest way would be to approximate this sum with an integral:
$$sum_{n=1}^kn^{-p}approx1+int_1^k x^{-p}dx=1+left.frac1{1-p}x^{1-p}rightrvert^k_0=1+frac{k^{1-p}}{1-p}.$$
answered Nov 16 at 7:33
YiFan
1,6891314
add a comment |
up vote
0
down vote
accepted
I am not aware of any closed form, but there are some good approximations. The easiest way would be to approximate this sum with an integral:
$$sum_{n=1}^kn^{-p}approx1+int_1^k x^{-p}dx=1+left.frac1{1-p}x^{1-p}rightrvert^k_0=1+frac{k^{1-p}}{1-p}.$$
answered Nov 16 at 7:33
YiFan
1,6891314
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
I am not aware of any closed form, but there are some good approximations. The easiest way would be to approximate this sum with an integral:
$$sum_{n=1}^kn^{-p}approx1+int_1^k x^{-p}dx=1+left.frac1{1-p}x^{1-p}rightrvert^k_0=1+frac{k^{1-p}}{1-p}.$$
answered Nov 16 at 7:33
YiFan
1,6891314
I am not aware of any closed form, but there are some good approximations. The easiest way would be to approximate this sum with an integral:
$$sum_{n=1}^kn^{-p}approx1+int_1^k x^{-p}dx=1+left.frac1{1-p}x^{1-p}rightrvert^k_0=1+frac{k^{1-p}}{1-p}.$$
answered Nov 16 at 7:33
YiFan
1,6891314
answered Nov 16 at 7:33
YiFan
1,6891314
answered Nov 16 at 7:33
YiFan
1,6891314
answered Nov 16 at 7:33
YiFan
1,6891314
1,6891314
add a comment |
add a comment |
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It may worth have a look at Hurwitz's zeta function.
– Kemono Chen
Nov 16 at 6:12
@KemonoChen: Hurwitz's zeta function is also infinite, check generalized harmonic numbers.
– Tianlalu
Nov 16 at 6:17
@Tianlalu You can make a subtraction from Riemann's zeta function.
– Kemono Chen
Nov 16 at 6:20
Thanks guys. I found the answer below helpful: math.stackexchange.com/questions/451558/…
– math_phile
Nov 16 at 6:45