Test the convergence of the series 1/1.2 - 1/3.4 + 1/5.6 - 1/7.8 + ... Infinity
Answers
The sequence un=1+1213+1231355+1233557 is being investigated to see if it is convergent.
The series' nth term is un=1233n135 as may be seen (2n1).
a) 135.5(2n1)=(2n)! a)\s2×4×6⋯×2n=(2n)!
2n×n!
⇒un=2n×(n!)
2(2n)\s⇒unun+1=2n×(n!)
b) 2n2n2n2n2n2n2n2n2n2n2n2n2n2n2 (2n)
2n+1×((n+1)!)
2=2n+1n+1.
Take, for example, the vn=1n2 sequence.
c) The nth term in the series is vn=1n2, as illustrated.
⇒vnvn+1=1n2×(n+1)21=n2+2n+1n2.\ss⇒unun+1−vnvn+1=2n+1n+1−n2+2n+1n2=n3−2n2−3n−1n2(n+1).
It's obvious that n32n23n1n2(n+1)>0 for n4.
Use unun+1>vnvn+1 for n4.
A positive term series 1np is convergent if and only if p>1.
A convergent value is vn=1n2.
d) If un and vn are two positive term series and unun+1vnvn+1, nm, there is a positive integer m.
If vn is convergent, then un is convergent, and vn is divergent otherwise. However, the series that is being offered is not complete.