Finding the Sum of a Product Series with a Given Upper Limit

[Justabeginner]

Homework Statement

Find Ʃ(product) with k=1 as the lower limit, and 50 as the upper limit. The formula is k/(k+2)

Homework Equations

The Attempt at a Solution

I noticed a pattern where the first few numbers are:
1/3, 2/4, 3/5, 4/6, 5/7 The denominator should cancel with the numerator of the next number in the pattern. I noticed this pattern for the odd numbered fractions. The last number in this odd numbered series will be 49/51, for which there are no cancellations possible, I think. For the even numbers, the pattern starts with 2/4 and ends with 50/52.

I think the fraction left will be (1*2)/(51*52)= 1/(51*26)

Is this logic right? I at first thought ..there is NO way they want me to multiply it out, and then realized there must be a pattern. So I tried and I hope I've got something. Thanks!

 

[Dick]

Homework Statement

Find Ʃ(product) with k=1 as the lower limit, and 50 as the upper limit. The formula is k/(k+2)

Homework Equations

The Attempt at a Solution

I noticed a pattern where the first few numbers are:
1/3, 2/4, 3/5, 4/6, 5/7 The denominator should cancel with the numerator of the next number in the pattern. I noticed this pattern for the odd numbered fractions. The last number in this odd numbered series will be 49/51, for which there are no cancellations possible, I think. For the even numbers, the pattern starts with 2/4 and ends with 50/52.

I think the fraction left will be (1*2)/(51*52)= 1/(51*26)

Is this logic right? I at first thought ..there is NO way they want me to multiply it out, and then realized there must be a pattern. So I tried and I hope I've got something. Thanks!

Yes, right.

 

[Justabeginner]

Thank you very much! I appreciate it :)

 

[Mark44]

Homework Statement

Find Ʃ(product) with k=1 as the lower limit, and 50 as the upper limit. The formula is k/(k+2)

The Attempt at a Solution

I noticed a pattern where the first few numbers are:
1/3, 2/4, 3/5, 4/6, 5/7 The denominator should cancel with the numerator of the next number in the pattern. I noticed this pattern for the odd numbered fractions. The last number in this odd numbered series will be 49/51, for which there are no cancellations possible, I think. For the even numbers, the pattern starts with 2/4 and ends with 50/52.

I think the fraction left will be (1*2)/(51*52)= 1/(51*26)

Is this logic right? I at first thought ..there is NO way they want me to multiply it out, and then realized there must be a pattern. So I tried and I hope I've got something. Thanks!

Your title is misleading, and your notation needs a little help. From your work, you aren't evaluating a sum, but a product. The notation for a product is a capital letter pi, or $\Pi$.

In LaTeX, the product would look like this:
$$ \prod_{k = 1}^{50}\frac{k}{k+2}$$
If you click the expression, you can see the LaTeX code that I wrote.

Upper case sigma (Ʃ) is used for sums. ∏ is used for products.

 

[Justabeginner]

Thank you. I did look at the LaTeX code, and I do understand how it is to be written on paper, though I just did not know how to write it with the coding. Note taken. :P

 

[Mark44]

\prod_{k = 1}^{50}\frac{k}{k+2}
This is the LaTeX code, which is written inside two pairs of $$ tags.
\prod makes the capital pi.
_ is used for subscripts or for the lower limit on integrals, sums, products, and so on.
^ is used for superscripts (exponents) or for the upper limit on integrals, sums, products, etc.
\frac writes the things in braces as a fraction.

 

[Justabeginner]

Oh wow, thank you so much!