Finding the Sum of a Series Using Partial Sums

In summary, the conversation discusses finding the Sn of a partial sum with the equation 1/n+3 - 1/n+1. The speaker attempted to find s1-s6 and concluded that the Sn is -1/n+2. Another speaker suggests using consecutive partial sums and subtracting them to find the Sn term directly.
  • #1
remaan
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0

Homework Statement



How to find the Sn of this patial sum : 1/n+3 - 1/ n+1 ??

Homework Equations



Finding the terms

The Attempt at a Solution


In fact, I tried finding s1 and s2 and so on till s6 and I found that the Sn is -1/ n+2 after I canceled the terms, is that right ??
 
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  • #2
If Sn= 1/(n+3)- 1/(n+1) then Sn+1= 1/(n+4)- 1/(n+2). Subtracting these two consecutive partial sums gives
[tex]]\frac{1}{n+4}- \frac{1}{n+2}- \frac{1}{n+3}+ \frac{1}{n+1}[/tex]
What does that give you?
 
  • #3
Are you trying to say that this method gives us Sn term directly, without a need for any subs. and finding terms ?
 
Last edited:

1. What are series (partial sums)?

Series (partial sums) refer to the sum of a sequence of numbers, where each term is added to the previous one. It is also known as a partial sum because it does not include all the terms of the sequence, but only a certain number of terms.

2. What is the formula for finding the partial sums of a series?

The formula for finding the partial sums of a series is Sn = a1 + a2 + a3 + ... + an, where Sn represents the sum of the first n terms in the series and a1, a2, a3, ... represent the individual terms of the series.

3. How do you determine if a series (partial sums) is convergent or divergent?

A series is convergent if the limit of its partial sums exists and is a finite number. In other words, if the partial sums approach a specific value as more terms are added, the series is convergent. On the other hand, if the partial sums do not approach a specific value and keep increasing or decreasing, the series is divergent.

4. What is the difference between an infinite series and a finite series?

A finite series is a series with a limited number of terms, while an infinite series has an infinite number of terms. In other words, a finite series has a definite sum, while an infinite series may or may not have a definite sum depending on whether it is convergent or divergent.

5. How can series (partial sums) be used in real-world applications?

Series (partial sums) have various applications in real-world problems, such as in calculating compound interest, estimating the value of a continuous function, and approximating the area under a curve. They are also used in financial analysis, statistics, and physics to model and analyze various phenomena.

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