Infinite Series (2)

kingwinner

Q1) Suppose {a_n} is a sequence of non-negative real numbers. Prove that
∞
∑  a_n converges if and only if
n=1
∞
∑  a_n / (1 + a_n) converges.
n=1

Q2) Determine whether the following series converges or diverges:
∞
∑  1/ln(e^n + e^-n)
n=1

請高手賜教, thx

[ 本帖最後由 kingwinner 於 10-3-22 03:56 PM 編輯 ]

Naozumi

另一題度好個暑假旅行計劃再想

kingwinner

Q1) 0<= a_n / (1+a_n) <1/2 for all n>N
=> 0<= a_n < (1+a_n)/2 for all n>N

i.e. 0<= a_n < 1 for all n>N ?
a_n/2 < a_n / (1+a_n) ?
呢兩行唔明...可以解釋下 ma?


good...暑假旅行計劃

[ 本帖最後由 kingwinner 於 10-3-23 08:45 AM 編輯 ]

Naozumi

唔係呀,做小小筆算你都懶??
a_n < (1+ a_n)/2
2a_n < 1 + an
a_n < 1

第二行自己諗

kingwinner

我有做筆算 ga, 但之前一版空白, 而家明了, thx

Naozumi