How do you find the power series for #f'(x)# and #int f(t)dt# from [0,x] given the function #f(x)=Sigma (n!)/n^nx^n# from #n=[1,oo)#?
2 Answers
Explanation:
The series for
From the FTC we know that:
It follows that the series for
Because
# f'(x) = sum_(n=1)^oo (n!)/n^(n-1) \ x^(n-1) #
# " " = 1 + x + 2/3x^2 + 3/8x^3 + 24/125x^4 + ... #
# int_0^x \ f(t) \ dt = sum_(n=1)^oo \ (n!)/n^n \ x^(n+1)/(n+1) #
# " " = 1/2x^2+ 1/6x^3 +1/18x^4+ 3/360x^5+... #
The radii of convergence for the original and subsequent series is
# |x| < 1 #
Explanation:
We have:
# f(x) = sum_(n=1)^oo \ (n!)/n^n \ x^n #
# " " = x + 1/2x^2 + 2/9x^3 + 3/32x^4 + 24/625x^5 + ... #
Let us also define the following functions:
# D(x) = f'(x) # and#I(x) = int_0^x \ f(t) \ dt #
Then:
# D(x) = d/dx sum_(n=1)^oo \ (n!)/n^n \ x^n #
# " " = sum_(n=1)^oo (n!)/n^n \ d/dx \ x^n #
# " " = sum_(n=1)^oo (n!)/n^n \ nx^(n-1) #
# " " = sum_(n=1)^oo (n!)/n^(n-1) \ x^(n-1) #
# " " = 1 + x + 2/3x^2 + 3/8x^3 + 24/125x^4 + ... #
And:
# I(x) = int_0^x \ sum_(n=1)^oo \ (n!)/n^n \ t^n \ dt #
# " " = sum_(n=1)^oo \ (n!)/n^n \ int_0^x \ \ t^n \ dt #
# " " = sum_(n=1)^oo \ (n!)/n^n \ [t^(n+1)/(n+1)]_0^x #
# " " = sum_(n=1)^oo \ (n!)/n^n \ x^(n+1)/(n+1) #
# " " = 1/2x^2+ 1/6x^3 +1/18x^4+ 3/360x^5+... #
For the radii of convergence, we can apply the d'Alembert's ratio test:
Suppose that;
# S=sum_(r=1)^oo a_n \ \ # , and#\ \ L=lim_(n rarr oo) |a_(n+1)/a_n| #
Then
if L < 1 then the series converges absolutely;
if L > 1 then the series is divergent;
if L = 1 or the limit fails to exist the test is inconclusive.
For the original series, the test limit is
# L = lim_(n rarr oo) | { ((n+1)!)/(n+1)^(n+1) \ x^(n+1) } / {(n!)/n^n \ x^n } | #
# \ \ = lim_(n rarr oo) | ((n+1)!)/(n+1)^(n+1) * n^n/(n!) * x^(n+1) / x^n | #
# \ \ = lim_(n rarr oo) | ((n+1))/(n+1)^(n+1) * n^n * x | #
# \ \ = lim_(n rarr oo) | 1/(n+1)^n * n^n * x | #
# \ \ = lim_(n rarr oo) | (n/(n+1))^n * x | #
# \ \ = lim_(n rarr oo) | (n/(n+1) * (1/n)/(1/n))^n * x | #
# \ \ = lim_(n rarr oo) | (1/(1+1/n))^n * x | #
# \ \ = | x | #
And so the series for
So for
# L = lim_(n rarr oo) | { ((n+1)!)/(n+1)^n \ x^n } / { (n!)/n^(n-1) \ x^(n-1) } | #
# \ \ = lim_(n rarr oo) | ((n+1)!)/(n+1)^n * n^(n-1)/(n!) * x^n/x^(n-1) | #
# \ \ = lim_(n rarr oo) | (n)/(n+1)^n * n^(n-1) * x | #
# \ \ = lim_(n rarr oo) | (n^n)/(n+1)^n * x | #
# \ \ = lim_(n rarr oo) | (n/(n+1))^n * x | #
# \ \ = lim_(n rarr oo) | (n/(n+1) * (1/n)/(1/n))^n * x | #
# \ \ = lim_(n rarr oo) | (1/(1+1/n))^n * x | #
# \ \ = | x | #
And so the series for
And finally, for
# L = lim_(n rarr oo) | { ((n+1)!)/ (n+1)^ (n+1) \ x^(n+2)/(n+2)} / {(n!)/n^n x^(n+1)/(n+1)} | #
# \ \ = lim_(n rarr oo) | ((n+1)!)/ (n+1)^ (n+1) * n^n/(n!) * x^(n+2)/(n+2) *(n+1)/x^(n+1)#
# \ \ = lim_(n rarr oo) | ((n+1))/ (n+1)^ (n+1) * n^n * x/(n+2) *(n+1)#
# \ \ = lim_(n rarr oo) | (1)/ (n+1)^n * n^n * x/(n+2) *(n+1) |#
# \ \ = lim_(n rarr oo) | (n/(n+1))^n * (n+1)/(n+2) * x | #
# \ \ = lim_(n rarr oo) | (n/(n+1) * (1/n)/(1/n))^n * (n+1)/(n+2) * (1/n)/(1/n)* x | #
# \ \ = lim_(n rarr oo) | (1/(1+1/n))^n * (1+1/n)/(1+2/n) * x | #
# \ \ = \ x | #
And so the series for
This should not come as any surprise as the radii of convergence is unaffected by differentiation or integration.