The Taylor Series of f(x) at the point is
Question 1. Let . Calculate the value of S.
Method (ii). Integrate the Taylor series of to show that S=1.
The Taylor series of is . Take the integration of the function on the interval [0,1], we get
The left hand side equals to 1 from integration by parts.
Method (iii). Differentiate the Taylor series of .
The Taylor series of is . Differentiate f(x) and get . Moreover, and .
Method (iv). Assume the function This implies f(0)=0. Assume
Since we get That means f(1)=1.
Method (v). Assume the function
Remark. There is a similar problem: calculate Answer is
Question 2. Let n be a positive integer. Prove that
and calculate the value of the summation
To calculate the value of S, there are two methods.
Method (i). The summation of S, n is only taken odd numbers. From the first step, we know the summation
Here we use the Taylor series of and .
Question 3. Assume
Question 4. Calculate the summation