13 -4 Infinite Sequences & Series
Rules:
lim
n (infinity)
(used to plug in large numbers)
1. if the degree of the top = the degree of the bottom then the answer is the coefficients
2. if the degree of the top is > the degree of the bottom = infinity
3. if the degree of the top is < the degree of the bottom is 0
* if the rules don't apply you will have to use your calculator to find what the sequence is approaching
* for geometric sequences if |r| < 1 then it goes to 0
Examples:
1. Rule #1
3n^3 + 5n^2 + 6n^4/2n^3 + 5n^4 = 6/5
2. Rule #2
lim cos (1/n)
n (infinity)
cos(1/100) = .99995
cos(1/1000) = 1
cos(1/10000) = 1
= 1
3. Rule #3
5n^2 + √n/3n^3 + 7 = 0
4. no rule
lim (-10)^n
n (infinity)
(-10)^100 = -10000
(-10)^1000 = -100000
= -(infinity)
13 - 5 Sum of Infinite Series
*can only be found when a geometric sequence where |r|<1
Formula: S = t1/1-r
*if |r| is not less than 1 we say the series diveges - doesn't approach a #
* if |r|<1 then we say the series approaches a #
Examples:
1. Find the sum of the infinite series: 9 - 6 + 4
(geometric)
r = -6/9 = -2
|-2/3| < 1
S = 9/(1 - (-2/3)) = 27/5
2. For what values of x does the series converges? : 1 + (x-2) + (x+2)^2 + (x-2)^3
|x-2/1| < 1
-1 < x < 1
1 < x < 3
ok i need help with the problems that involves sigma, so if anyone can help me with that that would be great...
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okay so i was kind of having trouble with this myself but i think i understand it now so i may be able to help a bit.
ReplyDeletealrighty so say you have this problem...
write the series expanded form.
the limits of summation are 4 and k=1. the index is k. and the summand is 5k.
to expand it, your answer would be 5+10+15+20.
*you have to have 4 numbers in the series becaue thats the number that is the limit of summation. your summand is 5k so you would multiply 1*5, 2*5, 3*5, 4*5 and your answer is 5+10+15+20.
i hope this helped a little but its hard to explain sigma notation on a computer. ill explain it more in class if you need me to!!!