Question

The nth term in the expansion of f(a+h) is


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Answer 1

Answer:

nCna^0h^n

Step-by-step explanation:

this is a bionomial expansuon question

Answer 2

$\displaystyle \sf The \: nth \: term \: in \: the \: expansion \: of \: f(a+h) \: is \: \: \frac{ {h}^{n - 1} }{(n - 1)!} {f}^{(n - 1)}(a)$

Given :

The function f(a + h)

To find :

The nth term in the expansion of f(a + h)

Solution :

Step 1 of 2 :

Find Taylor's Series expansion

If f(x + h) can be expanded as an infinite series then

$\displaystyle \sf f(x + h) = f(x) + hf'(x) + \frac{ {h}^{2} }{2!} f''(x) + \frac{ {h}^{3} }{3!} f'''(x) + . . . + \frac{ {h}^{n - 1} }{(n - 1)!} {f}^{(n - 1)}(x) + . . . . \infty$

f(x) possesses derivatives of all orders

Step 2 of 2 :

Find nth term in the expansion of f(a+h)

Putting x = a in the expansion of f(x + h) we get

$\displaystyle \sf f(a + h) = f(a) + hf'(a) + \frac{ {h}^{2} }{2!} f''(a) + \frac{ {h}^{3} }{3!} f'''(a) + . . . + \frac{ {h}^{n - 1} }{(n - 1)!} {f}^{(n - 1)}(a) + . . . . \infty$

Hence the nth term in the expansion of f(a+h)

$\displaystyle \sf{ = \frac{ {h}^{n - 1} }{(n - 1)!} {f}^{(n - 1)}(a) }$

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