Calculating or approximating $sum _{i=1}^{n} frac{1} {(a+(i-1)d)^i}$ and $sum _{i=1}^{n} frac{1-(i-1)alpha}...
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How it can be calculated or approximated the following partial sums:
$$S_n=sum _{i=1}^{n} frac{1} {(a+(i-1)d)^i},$$
$$S_n=sum _{i=1}^{n} frac{1-(i-1)alpha} {(a+(i-1)d)^i},$$
where $0<alpha<1$, $0<a<1$.
calculus sequences-and-series
edited Nov 18 at 9:22
Blue
47.1k870148
asked Nov 18 at 8:55
David
408
|
show 5 more comments
up vote
0
down vote
favorite
How it can be calculated or approximated the following partial sums:
$$S_n=sum _{i=1}^{n} frac{1} {(a+(i-1)d)^i},$$
$$S_n=sum _{i=1}^{n} frac{1-(i-1)alpha} {(a+(i-1)d)^i},$$
where $0<alpha<1$, $0<a<1$.
calculus sequences-and-series
edited Nov 18 at 9:22
Blue
47.1k870148
asked Nov 18 at 8:55
David
408
What's complex about it?
– José Carlos Santos
Nov 18 at 9:05
@JoséCarlosSantos I mean sophisticated.
– David
Nov 18 at 9:06
1
Look up en.m.wikipedia.org/wiki/Hurwitz_zeta_function and en.m.wikipedia.org/wiki/Polygamma_function
– Yuriy S
Nov 18 at 9:22
Sometimes there's no "nice" closed form. You can represent such sums as integrals, or approximate them for large or small parameters, etc. Mathematics is rich in opportunities
– Yuriy S
Nov 18 at 9:34
1
@David, summation index is not the argument. Please read carefully.
– Yuriy S
Nov 18 at 9:51
|
show 5 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
How it can be calculated or approximated the following partial sums:
$$S_n=sum _{i=1}^{n} frac{1} {(a+(i-1)d)^i},$$
$$S_n=sum _{i=1}^{n} frac{1-(i-1)alpha} {(a+(i-1)d)^i},$$
where $0<alpha<1$, $0<a<1$.
calculus sequences-and-series
edited Nov 18 at 9:22
Blue
47.1k870148
asked Nov 18 at 8:55
David
408
How it can be calculated or approximated the following partial sums:
$$S_n=sum _{i=1}^{n} frac{1} {(a+(i-1)d)^i},$$
$$S_n=sum _{i=1}^{n} frac{1-(i-1)alpha} {(a+(i-1)d)^i},$$
where $0<alpha<1$, $0<a<1$.
calculus sequences-and-series
calculus sequences-and-series
edited Nov 18 at 9:22
Blue
47.1k870148
asked Nov 18 at 8:55
David
408
edited Nov 18 at 9:22
Blue
47.1k870148
asked Nov 18 at 8:55
David
408
edited Nov 18 at 9:22
Blue
47.1k870148
edited Nov 18 at 9:22
Blue
47.1k870148
edited Nov 18 at 9:22
Blue
47.1k870148
47.1k870148
asked Nov 18 at 8:55
David
408
asked Nov 18 at 8:55
David
408
asked Nov 18 at 8:55
David
408
408
What's complex about it?
– José Carlos Santos
Nov 18 at 9:05
@JoséCarlosSantos I mean sophisticated.
– David
Nov 18 at 9:06
1
Look up en.m.wikipedia.org/wiki/Hurwitz_zeta_function and en.m.wikipedia.org/wiki/Polygamma_function
– Yuriy S
Nov 18 at 9:22
Sometimes there's no "nice" closed form. You can represent such sums as integrals, or approximate them for large or small parameters, etc. Mathematics is rich in opportunities
– Yuriy S
Nov 18 at 9:34
1
@David, summation index is not the argument. Please read carefully.
– Yuriy S
Nov 18 at 9:51
|
show 5 more comments
What's complex about it?
– José Carlos Santos
Nov 18 at 9:05
@JoséCarlosSantos I mean sophisticated.
– David
Nov 18 at 9:06
1
Look up en.m.wikipedia.org/wiki/Hurwitz_zeta_function and en.m.wikipedia.org/wiki/Polygamma_function
– Yuriy S
Nov 18 at 9:22
Sometimes there's no "nice" closed form. You can represent such sums as integrals, or approximate them for large or small parameters, etc. Mathematics is rich in opportunities
– Yuriy S
Nov 18 at 9:34
1
@David, summation index is not the argument. Please read carefully.
– Yuriy S
Nov 18 at 9:51
What's complex about it?
– José Carlos Santos
Nov 18 at 9:05
What's complex about it?
– José Carlos Santos
Nov 18 at 9:05
@JoséCarlosSantos I mean sophisticated.
– David
Nov 18 at 9:06
@JoséCarlosSantos I mean sophisticated.
– David
Nov 18 at 9:06
1
1
Look up en.m.wikipedia.org/wiki/Hurwitz_zeta_function and en.m.wikipedia.org/wiki/Polygamma_function
– Yuriy S
Nov 18 at 9:22
Look up en.m.wikipedia.org/wiki/Hurwitz_zeta_function and en.m.wikipedia.org/wiki/Polygamma_function
– Yuriy S
Nov 18 at 9:22
Sometimes there's no "nice" closed form. You can represent such sums as integrals, or approximate them for large or small parameters, etc. Mathematics is rich in opportunities
– Yuriy S
Nov 18 at 9:34
Sometimes there's no "nice" closed form. You can represent such sums as integrals, or approximate them for large or small parameters, etc. Mathematics is rich in opportunities
– Yuriy S
Nov 18 at 9:34
1
1
@David, summation index is not the argument. Please read carefully.
– Yuriy S
Nov 18 at 9:51
@David, summation index is not the argument. Please read carefully.
– Yuriy S
Nov 18 at 9:51
|
show 5 more comments
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What's complex about it?
– José Carlos Santos
Nov 18 at 9:05
@JoséCarlosSantos I mean sophisticated.
– David
Nov 18 at 9:06
1
Look up en.m.wikipedia.org/wiki/Hurwitz_zeta_function and en.m.wikipedia.org/wiki/Polygamma_function
– Yuriy S
Nov 18 at 9:22
Sometimes there's no "nice" closed form. You can represent such sums as integrals, or approximate them for large or small parameters, etc. Mathematics is rich in opportunities
– Yuriy S
Nov 18 at 9:34
1
@David, summation index is not the argument. Please read carefully.
– Yuriy S
Nov 18 at 9:51