- Level: International Baccalaureate
- Subject: Maths
- Word count: 1272
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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