The Taylor Series of f(x) at the point is

**Question 1. **Let . Calculate the value of S.

**Solution.**

**Method (i).**

**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

Therefore, f(1)=1.

**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

.

**Solution. **

.

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

.

**Method (ii).**

Since ,

.

Here we use the Taylor series of and .

**Question 3. **Assume

Prove

**Proof.**

**Question 4. **Calculate the summation

**Solution. **