Practice exam question on series

  • Thread starter Matt Armstrong
  • Start date
  • Tags
    Exam Series
In summary, the conversation discusses finding the value of the sum (infinity) Σ 2/((n+1)(n+3)), where n=1, using the integral test. It is mentioned that the partial sum formula is used, but the person is unsure of how to apply it and gets a result higher than the correct answer of 5/6. They are then advised to write the series as a telescoping series.
  • #1
Matt Armstrong

Homework Statement


Find the value of the sum

(Infinity)
Σ 2/((n+1)(n+3))
(n=1)

Homework Equations



Integral test

Partial Sum Formula = k/2 (a_1 + a_k)

The Attempt at a Solution



Admittedly I started off this problem the wrong way. I used the integral test thinking I might get an answer there, but only found that it converged, not that it was equal to what I got. A little puzzled at the usage of 'partial sum formula' but I do not recognize it. Looking through my book's index, 'partial sum' isn't even mentioned. I flipped through the relevant chapters and didn't see mention of it either.

But, it still confuses me as to how it works. If I plug in an arbitrarily large number, like 1 billion, I get well above the correct answer, which was marked as 5/6. How do I properly apply the Partial Sum formula?
 
Physics news on Phys.org
  • #2
Matt Armstrong said:

Homework Statement


Find the value of the sum

(Infinity)
Σ 2/((n+1)(n+3))
(n=1)

Homework Equations



Integral test

Partial Sum Formula = k/2 (a_1 + a_k)

The Attempt at a Solution



Admittedly I started off this problem the wrong way. I used the integral test thinking I might get an answer there, but only found that it converged, not that it was equal to what I got. A little puzzled at the usage of 'partial sum formula' but I do not recognize it. Looking through my book's index, 'partial sum' isn't even mentioned. I flipped through the relevant chapters and didn't see mention of it either.

But, it still confuses me as to how it works. If I plug in an arbitrarily large number, like 1 billion, I get well above the correct answer, which was marked as 5/6. How do I properly apply the Partial Sum formula?

Write it as a "telescoping series".
 

What is a series?

A series is a sequence of numbers or terms that are added together in a specific order. Each term in the series is called a "term", and the total number of terms in the series is called the "length" of the series.

What is the formula for finding the sum of a series?

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

How do you determine the convergence of a series?

A series is said to converge if the sum of its terms approaches a finite value as the number of terms increases. To determine the convergence of a series, you can use various tests such as the comparison test, the ratio test, or the integral test.

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

A finite series has a limited number of terms, while an infinite series has an infinite number of terms. In other words, a finite series has an end, while an infinite series does not. This also affects their convergence, as a finite series can either converge or diverge, while an infinite series can only converge or diverge to infinity.

How can series be applied in real life?

Series have various applications in real life, such as in finance, physics, and engineering. For example, compound interest can be represented as a series, and the motion of a falling object can be modeled using a series. Series can also be used in calculating probabilities and in data analysis.

Similar threads

How to show that these sequences are summable?
    • Calculus and Beyond Homework Help
Replies
1
Views
255
Convergence of the series nx^n
    • Calculus and Beyond Homework Help
Replies
2
Views
2K
Convergent Series Can Be Bounded by Any ##\epsilon>0##
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Is Using the Comparison Test to Prove Divergence Valid for This Series?
    • Calculus and Beyond Homework Help
Replies
29
Views
1K
Power Series and Convergence for ln(x+1)
    • Calculus and Beyond Homework Help
Replies
22
Views
3K
Finding an explicit formula for the sequence of partial sums
    • Calculus and Beyond Homework Help
Replies
2
Views
4K
How can a series be equal to this integral?
    • Calculus and Beyond Homework Help
Replies
14
Views
2K
Proving the convergence of series
    • Calculus and Beyond Homework Help
Replies
14
Views
1K
Prove divergence of the Series
    • Calculus and Beyond Homework Help
Replies
5
Views
1K
Derivation of the Fourier series of a real signal
    • Calculus and Beyond Homework Help
Replies
11
Views
1K

Hot Threads

Recent Insights

Back
Top