I have two problems that I need some help with. I believe I got to the answer but not quite sure if its right. I need to expand and evaluate each series.
1. There is an E like symbol that has a 7 at the top. On the bottom k=1 and on the right side (-1)^k K
2. There is an E like symbol that has a 8 at the top. ON the bottom K=3 and on the right side (-1)K/K(k-2)
∑ (capital sigma) = "the sum of" whatever follows.
The bottom number indicates where you start the sum, while the top indicates the end.
http://en.wikipedia.org/wiki/List_of_mathematical_symbols
Go down 3/4 of the way to the ∑ symbol.
I think you are describing sigma notation
∑ (-1)^k k from k-1 to 7
= (-1)^1 (1) + (-1)^2 (2) + (-1)^3 (3) + .... + (-1)^7 (7)
= -1 + 2 - 3 + 4 - ... -7
= -4
In the second I can't tell where the denominator ends
is it (-1)^ k /( (k(k-2)) ?
brackets are essential here
once you have decided on the proper meaning,
sub in k=3, 4, ..., 8 adding the resulting terms.
To expand and evaluate a series, we can follow a step-by-step process. Let's begin with the first series:
1. Expanding and evaluating the series with the symbol "E" like an uppercase sigma:
The series is denoted as:
7
E
k=1
(-1)^k * k
To expand the series, we substitute different values of k into the expression (-1)^k * k, starting with k=1, then k=2, and so on, until we reach the value of k given on the top (7 in this case).
Let's calculate the first few terms of the series:
For k=1, we have:
(-1)^1 * 1 = -1
For k=2, we have:
(-1)^2 * 2 = 2
For k=3, we have:
(-1)^3 * 3 = -3
For k=4, we have:
(-1)^4 * 4 = 4
For k=5, we have:
(-1)^5 * 5 = -5
For k=6, we have:
(-1)^6 * 6 = 6
For k=7, we have:
(-1)^7 * 7 = -7
Now, we can sum all the terms to evaluate the series:
-1 + 2 - 3 + 4 - 5 + 6 - 7
The sum is equal to -4.
Therefore, the expanded and evaluated series is -4.
Now let's move on to the second series:
2. Expanding and evaluating the series:
The series is denoted as:
8
E
k=3
(-1)^k / (k*(k-2))
To expand the series, we substitute different values of k into the expression (-1)^k / (k*(k-2)), starting with k=3, then k=4, and so on, until we reach the value of k given on the top (8 in this case).
Let's calculate the first few terms of the series:
For k=3, we have:
(-1)^3 / (3*(3-2)) = -1/3
For k=4, we have:
(-1)^4 / (4*(4-2)) = 1/8
For k=5, we have:
(-1)^5 / (5*(5-2)) = -1/15
For k=6, we have:
(-1)^6 / (6*(6-2)) = 1/24
For k=7, we have:
(-1)^7 / (7*(7-2)) = -1/35
For k=8, we have:
(-1)^8 / (8*(8-2)) = 1/48
Now, we can sum all the terms to evaluate the series:
-1/3 + 1/8 - 1/15 + 1/24 - 1/35 + 1/48
The sum is approximately equal to 0.002111.
Therefore, the expanded and evaluated series is approximately 0.002111.
I hope this helps! Let me know if you have any further questions.