Mister Exam
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- Product of series
-
How to use it?
- Sum of series:
- (-1)^n*x^n/(2*n+1)
- i^n
-
(1000!)/(n!*(1000-n)!)*((0.025)^n)*((1-0.025)^(1000-n))
- 7x^2
-
Identical expressions
- two /(one +(two n+ one)/ one thousand)^2
- 2 divide by (1 plus (2n plus 1) divide by 1000) squared
- two divide by (one plus (two n plus one) divide by one thousand) squared
- 2/(1+(2n+1)/1000)2
- 2/1+2n+1/10002
- 2/(1+(2n+1)/1000)²
- 2/(1+(2n+1)/1000) to the power of 2
- 2/1+2n+1/1000^2
- 2 divide by (1+(2n+1) divide by 1000)^2
-
Similar expressions
- 2/(1+(2n-1)/1000)^2
- 2/(1-(2n+1)/1000)^2
Sum of series 2/(1+(2n+1)/1000)^2
The solution
You have entered
[src]
oo _____ \ ` \ 2 \ -------------- \ 2 / / 2*n + 1\ / |1 + -------| / \ 1000 / /____, n = 1
$$\sum_{n=1}^{\infty} \frac{2}{\left(\frac{2 n + 1}{1000} + 1\right)^{2}}$$
Sum(2/(1 + (2*n + 1)/1000)^2, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{2}{\left(\frac{2 n + 1}{1000} + 1\right)^{2}}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = \frac{2}{\left(\frac{n}{500} + \frac{1001}{1000}\right)^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(\frac{n}{500} + \frac{1003}{1000}\right)^{2}}{\left(\frac{n}{500} + \frac{1001}{1000}\right)^{2}}\right)$$
Let's take the limit
we find
$$\frac{2}{\left(\frac{2 n + 1}{1000} + 1\right)^{2}}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = \frac{2}{\left(\frac{n}{500} + \frac{1001}{1000}\right)^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(\frac{n}{500} + \frac{1003}{1000}\right)^{2}}{\left(\frac{n}{500} + \frac{1001}{1000}\right)^{2}}\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
30601655624351528120014138182758711581530270957236018002637716626167436408997523361205453299413914278964368378950094270179649808638370458790842473237661148626792289603071010702366146505835875985394863363222801425397580687803436648271657182305376249744794206566354825428226484837126549398682301185625196542691206691530184713736703696066779014034559934461935563399557073722873873731822828626225661424893935260696791691669437492309855377827934661388555769413638902219236602334945185616166556875129045833557761937963681020300743960321612042421520489912898376811630187355783901824380171729850899612095507067898158646090695175869122698194192229904583851422674946059015598824946907136174248159550113159593130842558422080355499484502315029673271210896252191426228847461161832078613658703132137201362228367645321297343632202344679289763467299917817734409359909570441756288 2
- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + 250000*pi
12407402369220169096151194402588076289511667664642298023245960949949360607043008452538581602662133115436015813943843243726857488407931215485110091239344247329829023466961627954644690084586316768074368698853209502372375039524472645026976069958355149132368298915715225627216034279488058330156747593412663570006625429127451220416679546965657661608371466817707483788358616672311275173241475840302371561851098439183634420375937326463207909952067783526341558333242126478910251662244555171921891546448446010839467309775539874353809122479294304824102770588433632598512101447113244188183704237534290412709372564495544361558065034841112918863500328232076403281931916950451704245503297674313643166266623107617328269164359591017769334122524388905548205445020116077630957456910385336008772858608229064375982234855754998873370957692375080203303182726046517847034452753025
$$- \frac{30601655624351528120014138182758711581530270957236018002637716626167436408997523361205453299413914278964368378950094270179649808638370458790842473237661148626792289603071010702366146505835875985394863363222801425397580687803436648271657182305376249744794206566354825428226484837126549398682301185625196542691206691530184713736703696066779014034559934461935563399557073722873873731822828626225661424893935260696791691669437492309855377827934661388555769413638902219236602334945185616166556875129045833557761937963681020300743960321612042421520489912898376811630187355783901824380171729850899612095507067898158646090695175869122698194192229904583851422674946059015598824946907136174248159550113159593130842558422080355499484502315029673271210896252191426228847461161832078613658703132137201362228367645321297343632202344679289763467299917817734409359909570441756288}{12407402369220169096151194402588076289511667664642298023245960949949360607043008452538581602662133115436015813943843243726857488407931215485110091239344247329829023466961627954644690084586316768074368698853209502372375039524472645026976069958355149132368298915715225627216034279488058330156747593412663570006625429127451220416679546965657661608371466817707483788358616672311275173241475840302371561851098439183634420375937326463207909952067783526341558333242126478910251662244555171921891546448446010839467309775539874353809122479294304824102770588433632598512101447113244188183704237534290412709372564495544361558065034841112918863500328232076403281931916950451704245503297674313643166266623107617328269164359591017769334122524388905548205445020116077630957456910385336008772858608229064375982234855754998873370957692375080203303182726046517847034452753025} + 250000 \pi^{2}$$
-30601655624351528120014138182758711581530270957236018002637716626167436408997523361205453299413914278964368378950094270179649808638370458790842473237661148626792289603071010702366146505835875985394863363222801425397580687803436648271657182305376249744794206566354825428226484837126549398682301185625196542691206691530184713736703696066779014034559934461935563399557073722873873731822828626225661424893935260696791691669437492309855377827934661388555769413638902219236602334945185616166556875129045833557761937963681020300743960321612042421520489912898376811630187355783901824380171729850899612095507067898158646090695175869122698194192229904583851422674946059015598824946907136174248159550113159593130842558422080355499484502315029673271210896252191426228847461161832078613658703132137201362228367645321297343632202344679289763467299917817734409359909570441756288/12407402369220169096151194402588076289511667664642298023245960949949360607043008452538581602662133115436015813943843243726857488407931215485110091239344247329829023466961627954644690084586316768074368698853209502372375039524472645026976069958355149132368298915715225627216034279488058330156747593412663570006625429127451220416679546965657661608371466817707483788358616672311275173241475840302371561851098439183634420375937326463207909952067783526341558333242126478910251662244555171921891546448446010839467309775539874353809122479294304824102770588433632598512101447113244188183704237534290412709372564495544361558065034841112918863500328232076403281931916950451704245503297674313643166266623107617328269164359591017769334122524388905548205445020116077630957456910385336008772858608229064375982234855754998873370957692375080203303182726046517847034452753025 + 250000*pi^2
The graph
Examples of finding the sum of a series
- The Sum of the Power Series
x^n/n
(x-1)^n
- Factorial
1/2^(n!)
n^2/n!
x^n/n!
k!/(n!*(n+k)!)
- Flint Hills Series
csc(n)^2/n^3
- Basel problem
1/n^2
1/n^4
1/n^6
- Harmonic series
1/n
- Grandi's series
(-1)^n
- Alternating series
(-1)^(n + 1)/n
(n + 2)*(-1)^(n - 1)
(3*n - 1)/(-5)^n
(-1)^(n - 1)*n/(6*n - 5)
- Newton–Mercator series
(-1)^(n + 1)/n*x^n
- Exam the series for convergence
(3*n - 1)/(-5)^n