contestada

For the following function, find the Taylor series centered at x=7 and give the first 5 nonzero terms of the Taylor series. Write the interval of convergence of the series. f(x)=ln(x) f(x)= +∑n=1[infinity] f(x)= + + + + +⋯

The interval of convergence is: _________

Respuesta :

Answer:

[tex]ln(x) = log(7) + \frac{x-7}{7} - \frac{(x-7)^2}{98} + \frac{(x-7)^3}{1029} - \frac{(x-7)^4}{9604} + .....[/tex]

Step-by-step explanation:

Remember that the general formula for Taylor series is centered around  [tex]a[/tex]

is given by

[tex]f(x) = \sum\limits_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!} (x-a)^k[/tex]

For this case we have that   [tex]a = 7 , f(x) = ln(x)[/tex].  Therefore remember that

[tex]f(x)=ln(x)\\f'(x) = \frac{1}{x}\\f^{(2)}(x) = -\frac{1}{x^2}\\f^{(3)}(x) = \frac{2}{x^3}\\......[/tex]

Therefore

[tex]ln(x) = log(7) + \frac{x-7}{7} - \frac{(x-7)^2}{98} + \frac{(x-7)^3}{1029} - \frac{(x-7)^4}{9604} + .....[/tex]