Let P be an approximation of pai accurate to n decimals. Show that P + sin P gives an approximation correct to 3n decimals. (Hint: Let P = p + x.)

alesterp 2020-11-05 Answered
Let P be an approximation of pai accurate to n decimals. Show that P + sin P gives an approximation correct to 3n decimals.
(Hint: Let P=p + x.)
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Expert Answer

Mitchel Aguirre
Answered 2020-11-06 Author has 94 answers

Substitute the value of P  |f(x)|.
P=π + x
|f(x)|=|P + sin(P)  π|
=|π + x + sin(π + x)|
=|π + x + sin(π)cos x + cos π sin x  π|
=|x + 0 + (1)sinx|
=|x  sin x|
P is accurate to n decimals so,
|x|10n
10n x  10n
x [10n, 10n]
Use the Maclaurin Series of sin x expansion.
sin(x)=x  x33! + x55!
|f(x)|=|x  sin x|
=∣ x  (x  x33! + x55!)
=∣x33!  x55! < x33!
x33!  x55! < x33!
 < 10n33!
 < 103n

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