Subject: Calculus

# Infinite Series

## Calculus.InfiniteSeries History

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June 30, 2011
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Basic yet complex functions such as trigonometric and inverse trigonometric functions, exponential ~~and logarithmic ~~function, and rational functions can be understood well through infinite [[series]]. These functions can be simplified by expressing them in terms of infinite series in such a way that all of these functions can be approximated by polynomials, the simplest kind of function. These simpler functions can then be used to approximate more complex functions. Also, infinite series can show the close relationship between functions which seem to be quite different at first just like exponential and trigonometric functions.

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Basic yet complex functions such as trigonometric and inverse trigonometric functions, [[exponential function|exponential]] and [[natural logarithm|logarithmic function]], and rational functions can be understood well through infinite [[series]]. These functions can be simplified by expressing them in terms of infinite series in such a way that all of these functions can be approximated by polynomials, the simplest kind of function. These simpler functions can then be used to approximate more complex functions. Also, infinite series can show the close relationship between functions which seem to be quite different at first just like exponential and trigonometric functions.

June 30, 2011
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An infinite series is defined as the sum of infinitely many terms, such as

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An [[infinite series]] is defined as the sum of infinitely many terms, such as

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Basic yet complex functions such as trigonometric and inverse trigonometric functions, exponential and logarithmic function, and rational functions can be understood well through infinite series. These functions can be simplified by expressing them in terms of infinite series in such a way that all of these functions can be approximated by polynomials, the simplest kind of function. These simpler functions can then be used to approximate more complex functions. Also, infinite series can show the close relationship between functions which seem to be quite different at first just like exponential and trigonometric functions.

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Basic yet complex functions such as trigonometric and inverse trigonometric functions, exponential and logarithmic function, and rational functions can be understood well through infinite [[series]]. These functions can be simplified by expressing them in terms of infinite series in such a way that all of these functions can be approximated by polynomials, the simplest kind of function. These simpler functions can then be used to approximate more complex functions. Also, infinite series can show the close relationship between functions which seem to be quite different at first just like exponential and trigonometric functions.

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'''NEXT TOPIC''': [[Maclaurin series]]

March 06, 2011
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{$$\sum_{n \geq 1}^{\infty}a_n=\lim_{N \to \infty}\sum_{n}^{N}a_n$$}.

If the limit of such series exist in {$R$}, then we say that {$\sum_{n \geq 1}^{\infty}a_n$} is convergent, else, if the limit does not exist or is {$\pm infty$} then it is divergent.

If the limit of such series exist in {$R$}, then we say that {$\sum_{n \geq 1}^{\infty}a_n$} is convergent, else, if the limit does not exist or is {$\pm infty$} then it is divergent.

March 06, 2011
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Definition. Let {$a_1$}, {$a_2$}, {$a_3$} ,..., {$a_n$} be an infinite sequence of real numbers. The infinite series

~~∑~~_~~(~~n~~≥~~1~~)▒~~a_n is defined to be

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'''Definition.''' Let {$a_1$}, {$a_2$}, {$a_3$} ,..., {$a_n$} be an infinite sequence of real numbers. The infinite series {$\sum_{n \geq 1}^{\infty}a_n$} is defined to be

March 06, 2011
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Definition. Let ~~a1~~, ~~a2, ~~a~~ 3~~,~~…~~, be an infinite sequence of real numbers. The infinite series

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Definition. Let {$a_1$}, {$a_2$}, {$a_3$} ,..., {$a_n$} be an infinite sequence of real numbers. The infinite series

March 06, 2011
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Some irrational numbers like pi, ~~π~~ = 3.~~14159265358979…~~, can be expanded into a form of an infinite series as follows

~~π~~=3+ 1~~/~~10+ 4~~/〖~~10~~〗~~^2~~ ~~+~~ ~~1~~/〖~~10~~〗~~^3~~ ~~+~~ ~~5~~/〖~~10~~〗~~^4~~ ~~+~~ ~~9~~/〖~~10~~〗~~^5~~ ~~+~~ ~~2~~/〖~~10~~〗~~^6~~ ~~+~~ ~~6~~/〖~~10~~〗~~^7~~ ~~+~~ ~~5~~/〖~~10~~〗~~^8~~ ~~+3~~/〖~~10~~〗~~^9~~ ~~+ ~~…~~

This ~~in~~ ~~turn~~ ~~tells~~ ~~us~~ ~~that~~ ~~π~~ ~~is~~ ~~an~~ ~~infinite~~ ~~sum~~ ~~of fractions of this ~~form.

Importance:

This

Importance:

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Some irrational numbers like pi, {$\pi = 3.14159265358979...$}, can be expanded into a form of an infinite series as follows

{$$\pi =3+ \frac{1}{10}+ \frac{4}{10^2}+\frac{1}{10^3}+\frac{5}{10^4}+\frac{9}{10^5}+\frac{2}{10^6}+\frac{6}{10^7}+\frac{5}{10^8}+\frac{3}{10^9}+...$$}

This in turn tells us that {$\pi$} is an infinite sum of fractions of this form.

!!Importance:

{$$\pi =3+ \frac{1}{10}+ \frac{4}{10^2}+\frac{1}{10^3}+\frac{5}{10^4}+\frac{9}{10^5}+\frac{2}{10^6}+\frac{6}{10^7}+\frac{5}{10^8}+\frac{3}{10^9}+...$$}

This in turn tells us that {$\pi$} is an infinite sum of fractions of this form.

!!Importance:

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Convergent and Divergent Infinite Series

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!!Convergent and Divergent Infinite Series

March 06, 2011
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{$$1+ \frac{1}{3}+ \frac{1}{5}+\frac{1}{7}+ ... +\frac{1}{2n-1}+ ... = \~~sigma~~_{n=0}^{\infty}\frac{1}{2n-1}$$}

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{$$1+ \frac{1}{3}+ \frac{1}{5}+\frac{1}{7}+ ... +\frac{1}{2n-1}+ ... = \sum_{n=0}^{\infty}\frac{1}{2n-1}$$}

March 06, 2011
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{$$1+ \frac{1}{3}+ \frac{1}{5}+\frac{1}{7}+ ... +\frac{1}{2n-1}+ ... = \sigma_{n=0}^{\infty}\frac{1}{2n-1}

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{$$1+ \frac{1}{3}+ \frac{1}{5}+\frac{1}{7}+ ... +\frac{1}{2n-1}+ ... = \sigma_{n=0}^{\infty}\frac{1}{2n-1}$$}

March 06, 2011
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An infinite series is defined as the sum of infinitely many terms, such as

{$$1+ \frac{1}{3}+ \frac{1}{5}+\frac{1}{7}+ ... +\frac{1}{2n-1}+ ... = \sigma_{n=0}^{\infty}\frac{1}{2n-1}

Some irrational numbers like pi, π = 3.14159265358979…, can be expanded into a form of an infinite series as follows

π=3+ 1/10+ 4/〖10〗^2 + 1/〖10〗^3 + 5/〖10〗^4 + 9/〖10〗^5 + 2/〖10〗^6 + 6/〖10〗^7 + 5/〖10〗^8 +3/〖10〗^9 + …

This in turn tells us that π is an infinite sum of fractions of this form.

Importance:

Basic yet complex functions such as trigonometric and inverse trigonometric functions, exponential and logarithmic function, and rational functions can be understood well through infinite series. These functions can be simplified by expressing them in terms of infinite series in such a way that all of these functions can be approximated by polynomials, the simplest kind of function. These simpler functions can then be used to approximate more complex functions. Also, infinite series can show the close relationship between functions which seem to be quite different at first just like exponential and trigonometric functions.

Convergent and Divergent Infinite Series

Definition. Let a1, a2, a 3,…, be an infinite sequence of real numbers. The infinite series

∑_(n≥1)▒a_n is defined to be

∑_(n≥1)▒a_n = lim┬(N→∞)⁡∑_n^N▒a_n

If the limit of such series exist in R then we say that ∑_(n≥1)▒a_n is convergent, else, if the limit does not exist or is ±∞ then it is divergent.

Take note that we are not actually adding up all terms of the infinite series but we are adding up a finite number of terms and see their behavior as this terms approaches infinity. By this, we then define an infinite series as the limit of its sequence of partial sums.

{$$1+ \frac{1}{3}+ \frac{1}{5}+\frac{1}{7}+ ... +\frac{1}{2n-1}+ ... = \sigma_{n=0}^{\infty}\frac{1}{2n-1}

Some irrational numbers like pi, π = 3.14159265358979…, can be expanded into a form of an infinite series as follows

π=3+ 1/10+ 4/〖10〗^2 + 1/〖10〗^3 + 5/〖10〗^4 + 9/〖10〗^5 + 2/〖10〗^6 + 6/〖10〗^7 + 5/〖10〗^8 +3/〖10〗^9 + …

This in turn tells us that π is an infinite sum of fractions of this form.

Importance:

Basic yet complex functions such as trigonometric and inverse trigonometric functions, exponential and logarithmic function, and rational functions can be understood well through infinite series. These functions can be simplified by expressing them in terms of infinite series in such a way that all of these functions can be approximated by polynomials, the simplest kind of function. These simpler functions can then be used to approximate more complex functions. Also, infinite series can show the close relationship between functions which seem to be quite different at first just like exponential and trigonometric functions.

Convergent and Divergent Infinite Series

Definition. Let a1, a2, a 3,…, be an infinite sequence of real numbers. The infinite series

∑_(n≥1)▒a_n is defined to be

∑_(n≥1)▒a_n = lim┬(N→∞)⁡∑_n^N▒a_n

If the limit of such series exist in R then we say that ∑_(n≥1)▒a_n is convergent, else, if the limit does not exist or is ±∞ then it is divergent.

Take note that we are not actually adding up all terms of the infinite series but we are adding up a finite number of terms and see their behavior as this terms approaches infinity. By this, we then define an infinite series as the limit of its sequence of partial sums.