ch 10.3, 10.4, 11.1-11.6
Quiz yourself by thinking what should be in
each of the black spaces below before clicking
on it to display the answer.
Help!
|
|
||||
---|---|---|---|---|---|
What is the Geo Series? | diverges for absolute value of (r) >= 1
Converges for absolute value of (r)<1 to a sum of a/(1-r)
🗑
|
||||
What is the Test for Divergence | if the limit as n-->infinity of (a_n)= Does Not Exist OR limit as n-->infinity of (a_n) does not = 0 then the the Sum of (a_n) from n=1 to infinity is divergent
🗑
|
||||
Comparison Test | if both the Sum of a_n and the Sum of b_n are (+0 term then (a_n)>(b_n)>0 OR (a_n)>=(b_n)>=0
1) if the Sum of (a_n) converges then the Sum of (b_n) converges
2)if the Sum of (b_n) diverges then the Sum of (a_n) diverges
🗑
|
||||
slowest to quickest terms going to infinity | ln(n), n^a, a^n, n!, n^n
🗑
|
||||
Alternating Series Test | to use series MUST alternate every term
alternator: (-1)^n, (-1)^(n-1), cos(nPi)
(b_n)=the Absolute Value of (a_n)
if a)lim(b_n) as n->infinity=0 b) b_n is decreasing then the Sum of A_n is convergent
🗑
|
||||
Proof by induction | Specific case to general
a)Show decreasing between a_1 & a_2
b)assume sequence is decreasing with general terms for some a_(k+1)<a_k
c) show decreasing for the (k+2) case
🗑
|
||||
Integral Test | if f(x) is a function where f(n)=An for n>#
AND a)f cont for x>#
b)f is T term for x>=#
c)f is decr for x>=#
THEN a)Int from 1->infinity of f(x)dx Conv implies Sum of a_n from n=1->inf. conv
b)Int from 1->inf div impl. Sum n=1->inf a_n also div
🗑
|
||||
p-series | E n=2->inf 1/n^p<Int 1->inf 1/(x^p)dx<E n+1->inf 1/n^p
INT 1->inf. 1/(x^p)dx converges for p>1 and diverges for p=<1
🗑
|
||||
Limit Comparison Test | both a_n and b_n must be (+) term
Lim n->inf (a_n)/(b_n)
🗑
|
||||
Absolute/conditional convergence | E ABS V a_n
CONV
E(a_n) is abs conv
DIV
Test E(a_n)
*conv AST, cond conv.
*div E(a_n) div
🗑
|
||||
ratio Test | works well on x^n and n! always fails w/ only powers of n->1/n^2, 1/n^(1/2)
1)if lim n->inf AbsV (a_n+1)/a_n=L &L>1 then Sum A_n is Abs conv
2)if lim n->inf AbsV " =L &L<1 then Sum A_n is div
3)if " " =1 then test fails
🗑
|
||||
Root Test | use when nth power occurs ->n^n, a^n
1)if lim n->inf AbsV (a_n)^1/n= L<1 then Sum A_n is Abs Conv
2) if " " = L>1 then Sum a_n is div
3) if " " =1 then test fails
🗑
|
||||
Harmonic Series | Sum n=1->inf 1/n is divergent
🗑
|
||||
Telescoping Series | change Sum n=1->inf An to Sum i=1->n Ai solve using partial fractions, then write out 1st and last terms until things cancel. take lim n->inf of series terms
🗑
|
Review the information in the table. When you are ready to quiz yourself you can hide individual columns or the entire table. Then you can click on the empty cells to reveal the answer. Try to recall what will be displayed before clicking the empty cell.
To hide a column, click on the column name.
To hide the entire table, click on the "Hide All" button.
You may also shuffle the rows of the table by clicking on the "Shuffle" button.
Or sort by any of the columns using the down arrow next to any column heading.
If you know all the data on any row, you can temporarily remove it by tapping the trash can to the right of the row.
To hide a column, click on the column name.
To hide the entire table, click on the "Hide All" button.
You may also shuffle the rows of the table by clicking on the "Shuffle" button.
Or sort by any of the columns using the down arrow next to any column heading.
If you know all the data on any row, you can temporarily remove it by tapping the trash can to the right of the row.
Embed Code - If you would like this activity on your web page, copy the script below and paste it into your web page.
Normal Size Small Size show me how
Normal Size Small Size show me how
Created by:
dtbrowne7
Popular Math sets