The differentiation of the stated series expansion of f term-by-term to obtain the corresponding series expansion for the stated derivative of f is [tex]\sum_{n=1}^{\infty}[/tex] (-1)ⁿ(n3ⁿ)Xⁿ⁻¹
f(x) = 1/ (1 + 3x) = [tex]\sum_{n=0}^{\infty}[/tex] (-1)ⁿ3ⁿxⁿ
⇒ f'(x) = d/dx [[tex]\sum_{n=0}^{\infty}[/tex](-1)ⁿ 3ⁿXⁿ]
= [tex]\sum_{n=0}^{\infty}[/tex] (-1)ⁿ3ⁿ d/dx (xⁿ)
= [tex]\sum_{n=1}^{\infty}[/tex] (-1)ⁿ3ⁿ(nxⁿ⁻¹)
= [tex]\sum_{n=1}^{\infty}[/tex] (-1)ⁿ3ⁿnxⁿ⁻¹
= [tex]\sum_{n=1}^{\infty}[/tex] ((-1)ⁿnxⁿ⁻¹)3ⁿ
= [tex]\sum_{n=1}^{\infty}[/tex] (-1)ⁿ(n3ⁿ)xⁿ-1
In mathematics, differentiation is used to calculate rates of change. In mechanics, for example, velocity is the rate of change of displacement (with regard to time).
The acceleration is the rate of change of velocity (with respect to time).
They are employed in many fields, including
They can be used to represent:
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Full Question:
Differentiate the given series expansion of f term-by-term to obtain the corresponding series expansion for the derivative of f.
If f(x) = 1/ (1 + 3x) = [tex]\sum_{n=0}^{\infty}[/tex] (-1)ⁿ3ⁿxⁿ
