Answer
Verified
409.5k+ views
Hint: Power series is a series or sum of a sequence involving powers. The number of elements in the series can be finite or infinite. To represent the given function in the form of a power series take the help of infinite geometric series when $|r|<1$.
Complete step by step solution:
Many mathematical functions can be expressed or represented in the form of power series.Power series is a series or sum of a sequence involving powers. The number of elements in the series can be finite or infinite.A power series can be considered as a function of some variable (say x) . Suppose we have an infinite geometric series,
$S=1+r+{{r}^{2}}+{{r}^{3}}+.....$
This series can expressed with summation notation as,
$S=\sum\limits_{n=0}^{\infty }{{{r}^{n}}}$
We know that the above geometric series converges to $\dfrac{1}{1-r}$ when $|r|<1$. This means that when $|r|<1$,
$\sum\limits_{n=0}^{\infty }{{{r}^{n}}}=\dfrac{1}{1-r}$ ….. (i)
Now, let us consider the expression $\dfrac{1}{1-r}$ as a function by replacing r with x. Then equation (i) changes to $\dfrac{1}{1-x}=\sum\limits_{n=0}^{\infty }{{{x}^{n}}}$.
This means that $\dfrac{1}{1-x}=1+x+{{x}^{2}}+{{x}^{3}}+.....$ …. (ii)
Therefore, we found a function that can express or represent a power series and we also know that the above series is a converging series.Now, if we substitute the x as (-x) in equation (ii), then the equation will change in to
$\dfrac{1}{1-(-x)}=1+(-x)+{{(-x)}^{2}}+{{(-x)}^{3}}+.....$
$\Rightarrow \dfrac{1}{1+x}=1-x+{{x}^{2}}-{{x}^{3}}+{{x}^{4}}+.....$
Therefore, we represented the function $f(x)=\dfrac{1}{1+x}$ in the form of power series.If a converging series converges only when $|x|
Note: Some students may get confused between a converging series and a diverging series.A converging series is a series that has a finite value. Whereas a diverging series a series that does not have a finite value.Therefore, when $|r|>1$ the geometric series is a diverging series since it does not have a finite value.
Complete step by step solution:
Many mathematical functions can be expressed or represented in the form of power series.Power series is a series or sum of a sequence involving powers. The number of elements in the series can be finite or infinite.A power series can be considered as a function of some variable (say x) . Suppose we have an infinite geometric series,
$S=1+r+{{r}^{2}}+{{r}^{3}}+.....$
This series can expressed with summation notation as,
$S=\sum\limits_{n=0}^{\infty }{{{r}^{n}}}$
We know that the above geometric series converges to $\dfrac{1}{1-r}$ when $|r|<1$. This means that when $|r|<1$,
$\sum\limits_{n=0}^{\infty }{{{r}^{n}}}=\dfrac{1}{1-r}$ ….. (i)
Now, let us consider the expression $\dfrac{1}{1-r}$ as a function by replacing r with x. Then equation (i) changes to $\dfrac{1}{1-x}=\sum\limits_{n=0}^{\infty }{{{x}^{n}}}$.
This means that $\dfrac{1}{1-x}=1+x+{{x}^{2}}+{{x}^{3}}+.....$ …. (ii)
Therefore, we found a function that can express or represent a power series and we also know that the above series is a converging series.Now, if we substitute the x as (-x) in equation (ii), then the equation will change in to
$\dfrac{1}{1-(-x)}=1+(-x)+{{(-x)}^{2}}+{{(-x)}^{3}}+.....$
$\Rightarrow \dfrac{1}{1+x}=1-x+{{x}^{2}}-{{x}^{3}}+{{x}^{4}}+.....$
Therefore, we represented the function $f(x)=\dfrac{1}{1+x}$ in the form of power series.If a converging series converges only when $|x|
Note: Some students may get confused between a converging series and a diverging series.A converging series is a series that has a finite value. Whereas a diverging series a series that does not have a finite value.Therefore, when $|r|>1$ the geometric series is a diverging series since it does not have a finite value.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE