Question involving approximation, taylor series and proving Question: Consider

meli199939f

meli199939f

Answered question

2022-04-02

Question involving approximation, taylor series and proving
Question: Consider the approximation
ln(2)2(13+13×33+15×35)
Prove that the error in this approximation is less than
17×22×35
Attempt: It looks like the expression comes from the taylor series expansion so: ln(1+x)=xx22+x33x44+  for   1<x<1
ln(1x)=xx22x33x44+
ln(1+x1x)=2(x+x33+x55+x77)
Now let x=13
ln(2)=2(13+13×33+15×35+17×37)
So we have to prove that:
2(17×37+19×39+111×311)<17×22×35

Answer & Explanation

ostijum8dd

ostijum8dd

Beginner2022-04-03Added 7 answers

We can say that
2(17×37+19×39+111×311)<2(17×37+17×39+17×311)=2(17×37)÷(1-132)=17×22×35

Janessa Foster

Janessa Foster

Beginner2022-04-04Added 12 answers

Since 9>7, and 11>7, and 13>7, and so on, the tail
2737+2939+211311+213313+
is less than the sum of the geometric series
2737(1+132+134+136+).
But
1+132+134+136+=98.

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