1) The 1/2 is a constant, and so is pulled out front.

2) The integral of cos(x) is sin(x), by definition. 

3) Writing the limits for evaluation: 

4) Using the unit circle,   , and .

5)Simplifying:  

Infinite series can be added and subtracted with each other.

Since the 2 series are convergent, the sum of the convergent infinite series is also convergent.

Note: The starting value, in this case n=1, must be the same before adding infinite series together.

This is a fundamental property of series.

For any constant c, if  is convergent then  is convergent, and if  is divergent,  is divergent.

 is divergent in the question, and the constant c is 10 in this case, so  is also divergent.

A way to find out if the sum of the 2 infinite series is convergent or not is to find out whether the individual infinite series are convergent or not.

Test the first series 

.

This is a geometric series with .

By the geometric test, this series is convergent.

Test the second series 

.

This is a geometric series with .

By the geometric test, this series is convergent.

Since both of the series are convergent,  is also convergent. 

We can use the limit

to find the radius of convergence. We have

This means the radius of convergence is .

In order to figure if 

is convergent, divergent or neither, we need to use the ratio test.

Remember that the ratio test is as follows.

Suppose we have a series . We define,

Then if 

, the series is absolutely convergent.

, the series is divergent.

, the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply the ratio test to our problem.

Let  

and

Now 

.

Now lets simplify this expression to 

.

Since ,

we have sufficient evidence to conclude that the series is divergent.

This is an infinite geometric series.

The sum of an infinite geometric series can be calculated with the following formula,

 , where  is the first value of the summation, and r is the common ratio.

Solution:

Value of  can be found by setting 

r is the value contained in the exponent

This is an alternating series test.

In order to find the terms necessary to approximate the series within  first see if the series is convergent using the alternating series test. If the series converges, find n such that 

Step 1:

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

1.        
2.        {} is a decreasing sequence, or in other words 

Solution:

1.

2. {} is a decreasing functon, since a factorial never decreases.

Since the 2 tests pass, this series is convergent.

Step 2:

Plug in n values until 

4 needs to be added to approximate the sum within .

Step 1: Plug in values into the function and add up the fraction:



Step 2: Find the sum of the fractions....

We can convert the fractions to decimals:

Step 3: Round to  places...

Step 1: Define what an infinite sequence is...

An infinite sequence is a sequence that is non-terminating.

Step 2: Determine if each sequence above is infinite...

For , the sequence is always infinite because the set of factorials is infinite. Also, the set of values by raising two factorial powers together is also infinite, it never has an ending term.

For , this sequence is FINITE! 

For , this sequence is FINITE!