Some of the following problems involved modified known series. You will need to

Question

Some of the following problems involved modified known series. You will need to identify the modification and which series was used. Use the cubic approximation of e^x near x = 0 to estimate e^0.5. Use known series to find the series for the accumulation function integral^x_0 ln(1+t)dt around x = 0. Please note where the variable dependence x is, and please revise accumulation functions if you don't remember them. By recognizing known series, find the sum of the following convergent series: -1/2-(1/2)^2/2 - (1/2)^3/3 - (1/2)4/4 +... -5+5^2/2!-5^3/3!+5^4/4!-5^5/5!+... Solve the following series equation (i.e. find t): t^2-t^6/3!+t^10/5!-t^14/7!+...= squareroot 2/2.

Explanation / Answer

1. Cubic Approximation of a function of f(x) around zero

f3(x)=f(a)+f'(a) x + f''(a)(x-a)2/2! +f'''(a)(x-a)3/3!

f(x)=ex   f(0)=1

f'(x)=ex f'(0)=1

f''(x)=ex f''(0)=1

f'''(x)=ex f'''(0)=1

So ex = 1 +1.x + 1.x2/2 + 1.x3/3

For x=0.5

e0.5=1+1 x 0.5 + 1x0.52/2+ 1x0.53/3

= 1+0.5+0.125+0.04166

=1.666

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