# Infinite summation

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Introduction

Portfolio Task Infinite Summation ( Type 1 ) Gizem Özgören

11K

16/10/2008

1) Consider the following sequence of terms un , where

U0=

U1 =

U2 =

U3 =

.....

This question is about infinite summation, which is a way of expressing an infinite sum. The series consist of (n) terms where (n) effects the sum. The question challenges us to learn about series on ourselves and help us academically in further tasks / exams.

(i) Find an expression for un, in terms of n.

In this question, we are asked to find an expression for un , so what we have to do is to find a formula which may be applied to all the terms of the sequence and that will help us understand how (n) effect the terms.

First of all, we should start by giving the name un to the formula.

Un =

Middle

Now, we are asked to put in the n values according to the given domain ( and calculate the summation.

We should start with writing down all the terms that are suitable with the domain, calculate their value (using TI) and then calculate the summation. In addition, we should note that the answer should be in 6 decimal places, so the values of the terms should also be in 6 decimal places.

U0 =

U1 =

U2 =

U3 =

U4 =

U5 =

U6 =

U7 =

U8 =

U9 =

U10 =

Now, it is time to add all the terms up and in order to calculate accurately, I will use a TI.

Sn = u0 + u1 + u2 + u3 + u4 + u5+ u6 + u7 + u8 + u9 + u10

S10 =

(iii) Using technology, plot the relation between Sn and n.

Conclusion

n | Summation (Sn ) |

0 | 1 |

1 | 1.693147 |

2 | 1.933374 |

3 | 1.988878 |

4 | 1.998496 |

5 | 1.999829 |

6 | 3.540182 |

7 | 5.065455 |

8 | 6.387004 |

9 | 7.404813 |

10 | 14.459725 |

After calculating the summation and graphing it, I saw that the bigger n gets, so does Sn. The reason why is that the function grows exponentially and everytime you add up a new value, the summation increases. n and Sn directly proportional terms.

In conclusion, throughout the first question, I defined what an infinite summation is, found ways to express a series as a function, what permutation is and why we use it in order to create a function for the sequence, what the summation of the given sequence is and finally, how the factor n ( number of terms in the sequence ) effects the summation. I came to the answer that as n gets larger, by several interrogations such as calculating the values of each term, adding them up, graphing the data which lead met o the answer that in an infinite summation,

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

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