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

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In summary: I didn't realize that, and I am really glad you explained it to me that way. I will definitely be using this notation from now on!
  • #1
Justabeginner
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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!
 
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  • #2
Justabeginner said:

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.
 
  • #3
Thank you very much! I appreciate it :)
 
  • #4
Justabeginner said:

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.
 
  • #5
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
 
  • #6
\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.
 
  • #7
Oh wow, thank you so much!
 

1. What is summation notation or sigma notation?

Summation notation, also known as sigma notation, is a mathematical shorthand used to represent the sum of a series of numbers. It is denoted by the Greek letter sigma (Σ) followed by the expression to be summed, an index or variable, and the lower and upper limits of the sum.

2. How do you calculate the value of a summation using summation notation?

To calculate the value of a summation using summation notation, substitute the index or variable in the expression with each value within the given limits and add the resulting terms. For example, to calculate the value of Σx from x=1 to 5, you would substitute x with 1, 2, 3, 4, and 5 and add the resulting terms: 1+2+3+4+5=15.

3. What is the difference between summation notation and product notation?

Summation notation is used to represent the sum of a series of numbers, while product notation is used to represent the product of a series of numbers. Product notation is denoted by the capital letter pi (Π) followed by the expression to be multiplied, an index or variable, and the lower and upper limits of the product.

4. How do you calculate the value of a product using product notation?

To calculate the value of a product using product notation, substitute the index or variable in the expression with each value within the given limits and multiply the resulting terms. For example, to calculate the value of Πx from x=1 to 5, you would substitute x with 1, 2, 3, 4, and 5 and multiply the resulting terms: 1x2x3x4x5=120.

5. What are some common applications of summation and product notation in science?

Summation and product notation are commonly used in science to represent and calculate the total of a series of data points. They are particularly useful in statistics, physics, and chemistry, where large amounts of data are often summed or multiplied. For example, in physics, summation notation is used to calculate the total force on an object by summing the individual forces acting on it.

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