Calculus II Assignment # 5 
This assignment covers
Testing Infinite Series for Convergence
.
Questions
Test these infinite series for convergence.
State the test you use and if the series converges or diverges.
1)  2) 
3)  4) 
5)  6) 
7)  8) 
9)  10) 
11)  
.
Solutions
1) test for divergence

2) limit comparison test
since 9/5 > 1, the series diverges. 
3) basic comparison test Sb_{ n} = S (1 / 3^{ n} ) a convergent Geometric Series r = a = 1/3 so series converges. 
4) limit comparison test
we apply l'Hopital's rule to the limit 
5) test for divergence series diverges. 
6) since maximum value of cos n = 1, we know that Basic comparison test with a convergent pseries. Since 
7) reminder: e is a constant < 3.
so we apply l'Hopital's rule
we apply the rule again twice to get But e  3 < 0 so the fraction becomes so the series diverges. 
8) apply l'Hopital's rule. , l'Hopital again. , series diverges. ___________________________ 9) is a divergent pseries, p = ½ < 1 
10) use the integral test we set u = ln x and du = 1/x dx so the series converges.  
11) We will apply partial fraction decompostion for the Integral test.
so the series converges. 
.
(Back to Caculus II MathRoom Index)
.
(all content of the MathRoom Lessons © Tammy the Tutor; 2002  2005).