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# Solution for finding the sum of an infinite sequence

Extracts from this document...

Introduction

Internal Assessment 1

Solution for finding the sum of a infinite sequence

The objective of this assignment is to find out the sum of infinite sequences , where

In this equation,  is defined by the term number. For example,  is the first term, whereas  is the nth term.  are various variables. In this equation,  which is defined by . Likewise 3! is defined by 3, but there is an exception

First I will break down the equation so that it will be easier for me to find out he formula. I will examine the  defined as the sum of the first . For example, , . I am first going to use this equation , where , , and where  is . So it should look like this:

I will find out the  for .

In order to find , I used my TI-84 Plus to figure this out. I will plug in the equation                  , where the  is 1,  is 2 and  value is from 0 to 10. The method is shown in the appendix.

I came up with:

In order to check if I got it right, I used Microsoft Excel 2010. The method is shown in the appendix.

After seeing that my result matches the results in Excel, I decided to then find out the sum.

Middle

In order to find , I will only use Microsoft Excel 2010 since I know that it gave me the right result before. I will plug in the equation , where the  is 1, is 3 and  value is from 0 to 10. The method is shown in the appendix.

To find the results, I just subbed in 3 instead if . Now I will graph it using the same process:

By looking at this graph, I can say that when  value is past 10,  is 3 and it remains constant. This suggests that when  approaches  (infinity), sum of the infinity  value is 3.

After looking at the two sum, when, and when , I think that the general term is

Where  is the sum of the infinite numbers.  I came up with this formula because we know,

• For , when  kept increasing, the sum of the  value remained the same and that’s why . We can say that  instead of  because  can go up to  and on the other hand  is equivalent to 2.
• For , when  kept increasing, the sum of the  value remained the same and that’s why . We can say that  instead of  because  can go up to  and on the other hand  is equivalent to 3.

Thus,

Conclusion

After I pressed ENTER key, I got this result –

This suggest there is no result when . The reason behind this is that, when we punch in  or , we get the same error. As we cannot solve it.

Now I am going to let a,  and  in the equation .

I will use my TI-84 Plus in order to find out the situation. It looks like this –

So this means, when I use  as a negative number it will show some results. I will check if the result shown above different from the equation .

This means, that when  it is equal to when . But what happens when  is a negative number and  is a odd number. For example, when .

I will let ,  and  in  the equation .

I will again use my TI-84 Plus to find out the result –

This is what I get when I hit enter –

This simple means that you cannot solve a equation when there is a negative number. For example,

In conclusion, when  is a negative number,  had to be an even number to change the negative sign to positive sign.

In short, , ,  can be both positive and negative number,  can also be both negative and positive number but only when  is an even number.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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