Messing around with summation operators

In summary, the problem involves changing the operators in a summation and finding s_{2n}. The attempt at a solution is to rewrite s_{2n} as a summation using different values for the exponent. However, this is not necessary and the solution can simply be written as s_{2n}=\sum_{k=1} ^{2n} ((-1)^{k+1})/k. Additionally, the prompt reminds the person not to overthink the problem and to simply write s_{5} and s_{10} as summations.
  • #1
gcamilo
2
0

Homework Statement



I'm just not sure how to change the operators in summation, can anyone help?

Let [tex]s_{n}=\sum_{k=1} ^n ((-1)^{k+1})/k[/tex]

what is [tex]s_{2n}[/tex]?

Homework Equations



[tex]s_{n}=\sum_{k=1} ^n ((-1)^{k+1})/k[/tex]

The Attempt at a Solution



[tex]s_{2n}=\sum_{k=1} ^{2n} ((-1)^{2k+1})/2k[/tex]

or

[tex]s_{2n}=\sum_{k=1} ^{2n} ((-1)^{2k+2})/2k[/tex]

This is in order to figure out a series proof, I just think I am really naive.
 
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  • #2
gcamilo said:

Homework Statement



I'm just not sure how to change the operators in summation, can anyone help?

Let [tex]s_{n}=\sum_{k=1} ^n ((-1)^{k+1})/k[/tex]

what is [tex]s_{2n}[/tex]?


Homework Equations



[tex]s_{n}=\sum_{k=1} ^n ((-1)^{k+1})/k[/tex]

The Attempt at a Solution



[tex]s_{2n}=\sum_{k=1} ^{2n} ((-1)^{2k+1})/2k[/tex]
No
gcamilo said:
or

[tex]s_{2n}=\sum_{k=1} ^{2n} ((-1)^{2k+2})/2k[/tex]
No
gcamilo said:
This is in order to figure out a series proof, I just think I am really naive.

How would you write s5 as a summation (not expanded)? How about s10? Don't overthink this problem.
 
  • #3
Thanks! I guess I tried to overthink, I've just seen people do amazing things with summation sings and sequences, I just tried to imitate.
 

FAQ: Messing around with summation operators

What is a summation operator?

A summation operator is a mathematical symbol that represents the sum of a series of numbers or expressions. It is typically denoted by the symbol "Σ" and is used to indicate that the numbers or expressions within the series should be added together.

What is the purpose of a summation operator?

The purpose of a summation operator is to simplify the representation of a series of numbers or expressions that need to be added together. It allows for a more concise and efficient way of writing mathematical equations and is commonly used in various fields of science, such as physics and statistics.

How do you use a summation operator?

To use a summation operator, you first need to define the starting and ending values for the series. These are typically denoted as the lower and upper limits of the summation. Then, you can write the expression or formula to be summed in terms of the summation variable, which is often represented by the letter "i". Finally, you place the summation operator "Σ" in front of the expression, with the lower and upper limits written below and above the operator, respectively.

What are some common properties of summation operators?

Some common properties of summation operators include the commutative property, which states that the order of the terms being summed does not affect the result, and the distributive property, which allows for the distribution of a constant factor across the terms being summed. Additionally, summation operators follow the same rules of arithmetic, such as the associative and identity properties.

What are some applications of summation operators in science?

Summation operators are widely used in various fields of science, including physics, chemistry, and statistics. They are often used to calculate the total energy or mass of a system, to determine the average value of a series of measurements, or to represent a discrete probability distribution. They are also used in various mathematical models to simplify complex equations and make them more manageable for analysis and interpretation.

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