series

Question 1

partial sum:

Answer 1

the sum of the first n elements in a sequence, sn

Question 2

series:

Answer 2

a sequence of partial sums of a sequence
denoted with (n)Σ(k=1)ak, for a sequence (an)n

Question 3

if (n)Σ(k=1)ak converges:

Answer 3

(an)n is a null sequence

Question 4

if (an)n is a sequence with an>=0 for all n, these are equivalent:

Answer 4

(∞)Σ(n=1)an is convergent
the sequence of partial sums of (an)n is bounded

Question 5

riemanns rearrangement theorem:

Answer 5

let (an)n be a sequence such that (∞)Σ(n=1)an is convergent but (∞)Σ(n=1)|an| is divergent
for every real number r, there is a bijection f to and from the naturals such that (∞)Σ(n=1)a(f(n)) is convergent with (∞)Σ(n=1)a(f(n)) = r

Question 6

absolutely convergent:

Answer 6

(∞)Σ(n=1)an is absolutely convergent if (∞)Σ(n=1)|an| is convergent also
is conditionally convergent if not but that’s not important dw
every absolutely convergent series is convergent

Question 7

comparison test:

Answer 7

if (∞)Σ(n=1)an is absolutely convergent and (bn)n is a sequence with bn<=an for all n, then (∞)Σ(n=1)bn is absolutely convergent

Question 8

limit comparison test:

Answer 8

if an,bn>0 and (bn/an)n is convergent with lim(bn/an)!=0, then (∞)Σ(n=1)an is convergent <=> (∞)Σ(n=1)bn is convergent

Question 9

ratio test:

Answer 9

if (an)n is a sequence with an!=0 for all n, such that (|a(n+1)/a(n)|)n is convergent with lim(|a(n+1)/a(n)|) <1, then (∞)Σ(n=1)an is absolutely convergent