Series and Tests for Series Flashcards
Telescoping Series: Use definition (2) to see if a telescoping series converges or diverges.
Expand Sn , collapse the terms, and take the limit as n→∞
If the limit exists, then the series converges to this limit.
If the limit does not exist, the series diverges.


The converse of Theorem 6







Does a Harmonic Series Converge or Diverge?












1 / increasing function is
a decreasing function
The second method to prove a function is decreasing?
Show f’ (x) <0 eventually






A Useful Comparison for
n >3
is?
ln n >1
A Useful Comparison for a>0
is?
ln n < na
A Useful Comparison for
e 1/<em>n</em>
Is it is between?
1< e 1/<em>n </em>< e
A Useful Comparisons for tan-1
is it falls between
0-1 n < ∏/2 <2
A Useful Comparison for sin² x is it falls between?
0< sin² x < 1
A Useful Comparisons for cos2x
is it falls between?
0 < cos2x < 1












What is the 3 Step procedure when asked if a series is absolutely convergent, conditionally convergent, or diverges?
Step 1: Divergence test in your head on the positive part.
If the positive terms do not → 0, the series diverges by the DIVERGENCE TEST Stop.
If the terms → 0,, move to Step 2.
- *Step 2**: Check absolute convergence. If the series converges, it converges absolutely (and hence converges) and you are done. If no, move to Step 3.
- *Step 3**: Check for Conditional Convergence (with the alternating series test). If the series converges but not absolutely, then the series is conditionally convergent. We do not have any other resource in this course for non-positive series, so you must be given an alternating series if you have arrived at Step 3.



















