Can 1/r be expressed as a power series? [closed]

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Can 1/r be expressed as a power series, where r is a variable? Thank you! Also it would be helpful to learn the formal definition of a power series.

calculus sequences-and-series power-series

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edited Sep 1 at 21:21

Foobaz John

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asked Sep 1 at 21:17

Christina Daniel

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closed as off-topic by José Carlos Santos, Andrés E. Caicedo, Holo, Xander Henderson, Adrian Keister Sep 2 at 0:45

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, Holo, Xander Henderson, Adrian Keister

If this question can be reworded to fit the rules in the help center, please edit the question.

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  Re definitions, make sure to learn the distinction between these: en.wikipedia.org/wiki/Power_series en.wikipedia.org/wiki/Formal_power_series
  – J.G.
  Sep 1 at 21:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It can be represented as a power series in $x$: $$frac1r=sum_i=0^infty c_i x^i$$ where $c_0=1/r$ and $c_i=0$ for $ige1$.
  – AccidentalFourierTransform
  Sep 1 at 22:07
  

  
  
  
  

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up vote
0
down vote

favorite

2

Can 1/r be expressed as a power series, where r is a variable? Thank you! Also it would be helpful to learn the formal definition of a power series.

calculus sequences-and-series power-series

share|cite|improve this question

edited Sep 1 at 21:21

Foobaz John

18.6k41245

asked Sep 1 at 21:17

Christina Daniel

183

closed as off-topic by José Carlos Santos, Andrés E. Caicedo, Holo, Xander Henderson, Adrian Keister Sep 2 at 0:45

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, Holo, Xander Henderson, Adrian Keister

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Re definitions, make sure to learn the distinction between these: en.wikipedia.org/wiki/Power_series en.wikipedia.org/wiki/Formal_power_series
  – J.G.
  Sep 1 at 21:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It can be represented as a power series in $x$: $$frac1r=sum_i=0^infty c_i x^i$$ where $c_0=1/r$ and $c_i=0$ for $ige1$.
  – AccidentalFourierTransform
  Sep 1 at 22:07
  

  
  
  
  

add a comment | 

up vote
0
down vote

favorite

2

up vote
0
down vote

favorite

2

2

Can 1/r be expressed as a power series, where r is a variable? Thank you! Also it would be helpful to learn the formal definition of a power series.

calculus sequences-and-series power-series

share|cite|improve this question

edited Sep 1 at 21:21

Foobaz John

18.6k41245

asked Sep 1 at 21:17

Christina Daniel

183

Can 1/r be expressed as a power series, where r is a variable? Thank you! Also it would be helpful to learn the formal definition of a power series.

calculus sequences-and-series power-series

share|cite|improve this question

edited Sep 1 at 21:21

Foobaz John

18.6k41245

asked Sep 1 at 21:17

Christina Daniel

183

share|cite|improve this question

share|cite|improve this question

edited Sep 1 at 21:21

Foobaz John

18.6k41245

edited Sep 1 at 21:21

Foobaz John

18.6k41245

edited Sep 1 at 21:21

Foobaz John

18.6k41245

18.6k41245

asked Sep 1 at 21:17

Christina Daniel

183

asked Sep 1 at 21:17

Christina Daniel

183

asked Sep 1 at 21:17

Christina Daniel

183

183

closed as off-topic by José Carlos Santos, Andrés E. Caicedo, Holo, Xander Henderson, Adrian Keister Sep 2 at 0:45

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, Holo, Xander Henderson, Adrian Keister

If this question can be reworded to fit the rules in the help center, please edit the question.

closed as off-topic by José Carlos Santos, Andrés E. Caicedo, Holo, Xander Henderson, Adrian Keister Sep 2 at 0:45

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, Holo, Xander Henderson, Adrian Keister

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Re definitions, make sure to learn the distinction between these: en.wikipedia.org/wiki/Power_series en.wikipedia.org/wiki/Formal_power_series
  – J.G.
  Sep 1 at 21:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It can be represented as a power series in $x$: $$frac1r=sum_i=0^infty c_i x^i$$ where $c_0=1/r$ and $c_i=0$ for $ige1$.
  – AccidentalFourierTransform
  Sep 1 at 22:07
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Re definitions, make sure to learn the distinction between these: en.wikipedia.org/wiki/Power_series en.wikipedia.org/wiki/Formal_power_series
  – J.G.
  Sep 1 at 21:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It can be represented as a power series in $x$: $$frac1r=sum_i=0^infty c_i x^i$$ where $c_0=1/r$ and $c_i=0$ for $ige1$.
  – AccidentalFourierTransform
  Sep 1 at 22:07
  

  
  
  
  

1

1

Re definitions, make sure to learn the distinction between these: en.wikipedia.org/wiki/Power_series en.wikipedia.org/wiki/Formal_power_series
– J.G.
Sep 1 at 21:27

Re definitions, make sure to learn the distinction between these: en.wikipedia.org/wiki/Power_series en.wikipedia.org/wiki/Formal_power_series
– J.G.
Sep 1 at 21:27

It can be represented as a power series in $x$: $$frac1r=sum_i=0^infty c_i x^i$$ where $c_0=1/r$ and $c_i=0$ for $ige1$.
– AccidentalFourierTransform
Sep 1 at 22:07

It can be represented as a power series in $x$: $$frac1r=sum_i=0^infty c_i x^i$$ where $c_0=1/r$ and $c_i=0$ for $ige1$.
– AccidentalFourierTransform
Sep 1 at 22:07

add a comment | 

2 Answers
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up vote
8
down vote



accepted

Write
$$
frac1r=frac11-(1-r)=sum_n=0^infty (1-r)^n
$$
for $|r-1|<1$ by the geometric series.

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answered Sep 1 at 21:21

Foobaz John

18.6k41245

add a comment | 

up vote
6
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To answer the second part of your question, a power series is an expression of the form
$$
c_0 + c_1(x-a) + c_2(x-a)^2 + c_3 (x-a)^3 + dots = sum_n=0^infty c_n (x-a)^n
$$
where $a$, $c_0, c_1, c_2, dots$ are constants, and $x$ is a variable. We say this power series is centered at $a$. You can think of it as an infinite polynomial. Indeed, if you cut off the series at any finite number of terms, you have a polynomial.

When you ask if $frac1r$ can be expressed as a power series, that's not quite a complete question. As Foobaz John shows, you can represent $frac1r$ as a power series centered at $1$. You can't express $frac1r$ as a power series centered at $0$, though, mainly because $frac1r$ blows up at $0$.

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answered Sep 1 at 21:29

Matthew Leingang

15.6k12144

add a comment | 

2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
8
down vote



accepted

Write
$$
frac1r=frac11-(1-r)=sum_n=0^infty (1-r)^n
$$
for $|r-1|<1$ by the geometric series.

share|cite|improve this answer

answered Sep 1 at 21:21

Foobaz John

18.6k41245

add a comment | 

up vote
8
down vote



accepted

Write
$$
frac1r=frac11-(1-r)=sum_n=0^infty (1-r)^n
$$
for $|r-1|<1$ by the geometric series.

share|cite|improve this answer

answered Sep 1 at 21:21

Foobaz John

18.6k41245

add a comment | 

up vote
8
down vote



accepted

up vote
8
down vote



accepted

Write
$$
frac1r=frac11-(1-r)=sum_n=0^infty (1-r)^n
$$
for $|r-1|<1$ by the geometric series.

share|cite|improve this answer

answered Sep 1 at 21:21

Foobaz John

18.6k41245

Write
$$
frac1r=frac11-(1-r)=sum_n=0^infty (1-r)^n
$$
for $|r-1|<1$ by the geometric series.

share|cite|improve this answer

answered Sep 1 at 21:21

Foobaz John

18.6k41245

share|cite|improve this answer

share|cite|improve this answer

answered Sep 1 at 21:21

Foobaz John

18.6k41245

answered Sep 1 at 21:21

Foobaz John

18.6k41245

answered Sep 1 at 21:21

Foobaz John

18.6k41245

18.6k41245

add a comment | 

add a comment | 

up vote
6
down vote

To answer the second part of your question, a power series is an expression of the form
$$
c_0 + c_1(x-a) + c_2(x-a)^2 + c_3 (x-a)^3 + dots = sum_n=0^infty c_n (x-a)^n
$$
where $a$, $c_0, c_1, c_2, dots$ are constants, and $x$ is a variable. We say this power series is centered at $a$. You can think of it as an infinite polynomial. Indeed, if you cut off the series at any finite number of terms, you have a polynomial.

When you ask if $frac1r$ can be expressed as a power series, that's not quite a complete question. As Foobaz John shows, you can represent $frac1r$ as a power series centered at $1$. You can't express $frac1r$ as a power series centered at $0$, though, mainly because $frac1r$ blows up at $0$.

share|cite|improve this answer

answered Sep 1 at 21:29

Matthew Leingang

15.6k12144

add a comment | 

up vote
6
down vote

To answer the second part of your question, a power series is an expression of the form
$$
c_0 + c_1(x-a) + c_2(x-a)^2 + c_3 (x-a)^3 + dots = sum_n=0^infty c_n (x-a)^n
$$
where $a$, $c_0, c_1, c_2, dots$ are constants, and $x$ is a variable. We say this power series is centered at $a$. You can think of it as an infinite polynomial. Indeed, if you cut off the series at any finite number of terms, you have a polynomial.

When you ask if $frac1r$ can be expressed as a power series, that's not quite a complete question. As Foobaz John shows, you can represent $frac1r$ as a power series centered at $1$. You can't express $frac1r$ as a power series centered at $0$, though, mainly because $frac1r$ blows up at $0$.

share|cite|improve this answer

answered Sep 1 at 21:29

Matthew Leingang

15.6k12144

add a comment | 

up vote
6
down vote

up vote
6
down vote

To answer the second part of your question, a power series is an expression of the form
$$
c_0 + c_1(x-a) + c_2(x-a)^2 + c_3 (x-a)^3 + dots = sum_n=0^infty c_n (x-a)^n
$$
where $a$, $c_0, c_1, c_2, dots$ are constants, and $x$ is a variable. We say this power series is centered at $a$. You can think of it as an infinite polynomial. Indeed, if you cut off the series at any finite number of terms, you have a polynomial.

When you ask if $frac1r$ can be expressed as a power series, that's not quite a complete question. As Foobaz John shows, you can represent $frac1r$ as a power series centered at $1$. You can't express $frac1r$ as a power series centered at $0$, though, mainly because $frac1r$ blows up at $0$.

share|cite|improve this answer

answered Sep 1 at 21:29

Matthew Leingang

15.6k12144

To answer the second part of your question, a power series is an expression of the form
$$
c_0 + c_1(x-a) + c_2(x-a)^2 + c_3 (x-a)^3 + dots = sum_n=0^infty c_n (x-a)^n
$$
where $a$, $c_0, c_1, c_2, dots$ are constants, and $x$ is a variable. We say this power series is centered at $a$. You can think of it as an infinite polynomial. Indeed, if you cut off the series at any finite number of terms, you have a polynomial.

When you ask if $frac1r$ can be expressed as a power series, that's not quite a complete question. As Foobaz John shows, you can represent $frac1r$ as a power series centered at $1$. You can't express $frac1r$ as a power series centered at $0$, though, mainly because $frac1r$ blows up at $0$.

share|cite|improve this answer

answered Sep 1 at 21:29

Matthew Leingang

15.6k12144

share|cite|improve this answer

share|cite|improve this answer

answered Sep 1 at 21:29

Matthew Leingang

15.6k12144

answered Sep 1 at 21:29

Matthew Leingang

15.6k12144

answered Sep 1 at 21:29

Matthew Leingang

15.6k12144

15.6k12144

add a comment | 

add a comment |