Question

[Hint. Use Taylor's series]
х
ot
(ii) Prove that log(x + h) = logh +
22
2h2
+
h
3h​

Answer 1

Answer: Proof of the equation of the series expansion of log(x+h) is as follows that :-

Prove that log(x+h)=logh+(x/h)-x²/2h²+x³/3h³+....... upto infinity

Explanation:

[Hint. Use Taylor's series]

х

ot

(ii) Prove that log(x + h) = logh +

22 2 h2 + h3h​:-

Using the Taylor expansion the first thing is that the formula contains the derivatives in increasing order and finally putting all the values in the formula at x=0 in:-

Here f(x)=logh at x=0,

and derivative of f(x)= 1/x+h at x=0 it is 1/h.

and then double derivative is -1/(x+h)² at x=0 is -1/h²

and finally the triple derivatie is 2/(x+h)³ at x=0 is 2/h³

and the expansion goes on till infinity.

Here using the taylor's series now:-

f(x)=f(0)+(summation of nth derivative/factorial n)x^n,

Using this formule and putting all the values of the derivative :-

The expansion is log(x+h)=logh(at x=0)+(x/h)-x²/2h²+2x³/6h³ the expansion is:-

log(x+h)=logh(at x=0)+(x/h)-x²/2h²+x³/3h³+....... upto infinity

Hence proving the result.

Thus proved.

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