Infinite Series problem? ?

Question

sin(x)= x-(1/3!)x^3 + (1/5!)x^5 - (1/7!)x^7 + (1/9!)x^9 - (1/11!)x^11 + (1/13!)x^13 - (1/15!)x^15 ...

Use the first 7 terms to estimate sin(1). Carry all decimals.

Compare that estimate to what your calculator produces.

Do the same for sin (3) and sin (5).

How does the accuracy of the series change as the x value grows?

Differentiate sin x and its series. What have you found?

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(If you can, can you also please post steps on how to differentiate the series. I have no idea how to.) Any help would be greatly appreciated. Thanks!

Answer 1

a)  Please remember that sin(1) means 1 RADIAN on your calculator.

     you could use pi/3 radians in that formula but not corresponding 60 degrees.

b)  differentiating sin x yields cos x

differentiating each term of the sin series by the roll down method yields the following:

1 - (1/2!)x^2  + (1/4!)x^4 -  (1/6!)x^6 + ...  which is the series for cos x

Hope that helps