The question:Rewriting Series with Sigma Notation

In summary, the conversation revolved around the topic of sigma and series notation, specifically in relation to a given question. The individual had provided their answer using incorrect notation, which was pointed out and corrected by others in the conversation. The importance of using parentheses in mathematical expressions was also discussed.
  • #1
Mo
81
0
Series and "Sigma" Notation

I have been revising over the sigma/sequences and series chapters, this is the second question now where i have had different answers to the book - yet- my answers seem to work - i think...

The question :

Write in [tex]\sum[/tex] notation

1 - 2 + 4 - 8 + 16 - 32


My answer is:

[tex]\sum_{0}^5 -2^r[/tex]

Is this correct?

Thie answer in the book by the way is:

[tex]\sum_{1}^6 (-1)^{r+1} \ 2r^{r-1}[/tex]

Regards,
Mo
 
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  • #2
Mo said:
I have been revising over the sigma/sequences and series chapters, this is the second question now where i have had different answers to the book - yet- my answers seem to work - i think...

The question :

Write in [tex]\sum[/tex] notation

1 - 2 + 4 - 8 + 16 - 32


My answer is:

[tex]\sum_{0}^5 -2^r[/tex]

Is this correct?
...

Nope. Every term in that sum is negative (which is not true for 1 - 2 + 4...).
 
  • #3
Then the answer in the books seems wrong and yours as well.
[tex] 1-2+4-8+16-32=(-)^{0}2^{0}+(-)^{1}2^{1}+(-)^{2}2^{2}+(-)^{3}2^{3}+(-)^{4}2^{4}+(-)^{5}2^{5}=\sum_{k=0}^{5}(-)^{k}2^{k} [/tex]

You might have mistyped the answer in the book.

Daniel.
 
  • #4
Yes i have typed in ther answer from the book wrongly, very sorry about that.

[tex]\sum_{1}^6 (-1)^{r+1} \ 2^{r-1}[/tex]

is the correct one.

However i still can't see how my answer is wrong!

when r is 0 , the answer is +1
when r is 1 , the answer is -2
when r is 2 , the answer is +4
when r is 3 , the answer is -8

so this would mean +1-2+4-8 ? Or maybe I am making a really stupid mistake here!

Thanks for your replies so far!
 
  • #5
Mo said:
However i still can't see how my answer is wrong!

when r is 0 , the answer is +1
when r is 1 , the answer is -2
when r is 2 , the answer is +4
when r is 3 , the answer is -8

so this would mean +1-2+4-8 ? Or maybe I am making a really stupid mistake here!

Thanks for your replies so far!

At first, I thought your answer was right. It's not, cos your sum is -2^r and not (-2)^r. When r is 0, for your answer, you get -1, ie. -1 x 2^0.
 
  • #6
Your answer would have been correct if you would have used this one: -

The following code was used to generate this LaTeX image:



[tex] \sum_{k=0}^{5}(-2)^{k} [/tex]
 
  • #7
Look at it this way: If you had

[tex]\sum (1-2^r)[/tex]

would you say that was

[tex](1-1) + (1-2) + (1-4) + ...[/tex]

or

[tex](1+1) + (1-2) + (1+4) + ...[/tex]

?

When you evaluate an expression that doesn't have parentheses inside of it, and is on a single line, exponents always come first, then multiplication/division, then addition/subtraction.
 
  • #8
Thank you for your replies all

Offcourse i should have used brackets :sigh: next time ill remember!

thanks again

Regards,
Mo

PS: JTbell, this first one ..
 
  • #9
I just realized that I got the signs backwards on my second choice.

Oh well, you got the right idea, anyway!
 

Related to The question:Rewriting Series with Sigma Notation

1. What is a series in mathematics?

A series in mathematics is a sum of a sequence of numbers, where the terms are added in a specific order. It can be finite, with a specific number of terms, or infinite, with an infinite number of terms.

2. What is sigma notation?

Sigma notation is a shorthand way of writing a series. It uses the Greek letter sigma (Σ) to represent the sum of the terms in a series. It is written as Σ an, where n is the index of the terms and an is the nth term of the series.

3. What is the purpose of using sigma notation?

The purpose of using sigma notation is to simplify and condense the expression of a series. It is also useful for representing infinite series, as it allows us to write the general term of the series and the limits of the series in a compact form.

4. How do you evaluate a series using sigma notation?

To evaluate a series using sigma notation, you need to substitute the values of the index n into the general term an and add up all the terms. The result is the sum of the series.

5. What is the difference between arithmetic and geometric series?

Arithmetic series is a series in which the difference between two consecutive terms is constant. Geometric series is a series in which the ratio between two consecutive terms is constant. In an arithmetic series, the terms increase or decrease by a fixed amount, while in a geometric series, the terms increase or decrease by a fixed ratio.

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