Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store
seo-qna
SearchIcon
banner

What is the radius of convergence of the Maclaurin’s series expansion for the function f(x)=sinhx?

Answer
VerifiedVerified
444.3k+ views
like imagedislike image
Hint: We solve this problem by finding the Maclaurin’s expansion of the given function. The Maclaurin;s expansion of a function p(x) is given as,
p(x)=p(0)+x1!p(0)+x22!p(0)+x33!p(0)+........
We use the standard formula of hyperbolic sine trigonometric function that is,
sinhx=exex2
After finding the Maclaurin’s series expansion we find the values of x for which the series is convergent using the ratio test which is given as if an is the nth term of the series and,
l=limn|an+1an|
If l<1 then the series is convergent and if l1 then the series is divergent.

Complete step-by-step solution:
We are given that the function as,
f(x)=sinhx
We know that the formula of hyperbolic sine function is given as,
sinhx=exex2
By using this formula we get the given function as,
f(x)=exex2
Now, let us find the Maclaurin’s expansion for the above function.
Let us substitute x=0 in above function then,
f(0)=e0e(0)2=0
Now, let us differentiate the given function and substitute x=0 then we get,
f(x)=ex+ex2f(0)=e0+e02=1
Now, let us differentiate the function again and substitute x=0 then we get,
f(x)=exex2f(0)=e0e02=0
Now, let us differentiate the function again and substitute x=0 then we get,
f(x)=ex+ex2f(0)=e0+e02=1
We know that the Maclaurin’s expansion for a function p(x) is given as,
p(x)=p(0)+x1!p(0)+x22!p(0)+x33!p(0)+........
By using this formula to above given function then we get the Maclaurin expansion as,
f(x)=0+x1!(1)+x22!(0)+x33!(1)+........f(x)=x+x33!+x55!+......
Now, let us represent the above series in summation then we get,
f(x)=n=0x2n+1(2n+1)!
Here, we can see that the nth term of above series is given as,
an=x2n+1(2n+1)!
Now, let us use the ratio test.
We know that the ratio test which is given as if an is the nth term of the series and,
l=limn|an+1an|
If l<1 then the series is convergent and if l1 then the series is divergent.
Now, let us find the value of l for the given function then we get,
l=limn|[x2(n+1)+1(2(n+1)+1)!][x2n+1(2n+1)!]|l=limn|x2n+3(2n+3)!×(2n+1)!x2n+1|
Here, we know that the term (2n+3)! can be represented in terms of (2n+1)! as,
(2n+3)!=[(2n+3)(2n+2)](2n+1)!
By replacing this value in above equation then we get,
l=limn|x2n+1×x2[(2n+3)(2n+2)](2n+1)!×(2n+1)!x2n+1|l=|x2|limn|1(2n+3)(2n+2)|
Here, we can see that, as the value n the denominator in the limit tends to so that fraction inside the limit tends to ‘0’.
So, we can take the value of l as,
l=0<1
Here, we can say that the given function is convergent.
So, we can say that for any value of x the series is convergent.
We know that the radius of convergence is given as the value of x for which the function is convergent.
So, the radius of convergence is given as real numbers R
Therefore, we can conclude that the radius of given function f(x)=sinhx is given as,
xR


Note: Students may make mistakes by not giving the radius of convergence.
To find the radius of convergence we need to find the value of x where the given function or series is convergent so that the answer for this problem should be xR
But students may make mistakes by concluding that the given function is convergent. Given function may or may not be convergent for any value of x so we need to find those values. Bit concluding the question directly mentioning that the given function is convergent is not the correct answer for the question.
We need to give the answer which is relevant to the question asked.

Latest Vedantu courses for you
Grade 10 | CBSE | SCHOOL | English
Vedantu 10 CBSE Pro Course - (2025-26)
calendar iconAcademic year 2025-26
language iconENGLISH
book iconUnlimited access till final school exam
tick
School Full course for CBSE students
PhysicsPhysics
Social scienceSocial science
ChemistryChemistry
MathsMaths
BiologyBiology
EnglishEnglish
₹41,000 (9% Off)
₹37,300 per year
Select and buy