[58] For a nonnegative term series Σa_n, if there exists l ∈ R ∪ {+∞} s.t. a_{n+1}/a_n → l (n → ∞), then Σa_n converges if l ＜ 1, and Σa_n diverges if l ＞ 1.

https://www.evernote.com/shard/s29/sh/6d93b03e-5842-4301-bf2e-64574de00c33/3ead698a49bac72999dcc8d6f81a27dc

[57] For any nonnegative term series Σa_n, [0≦∃k＜1. ∃m∈N. ∀n∈N. n≧m ⇒ (n√a_n ≦ k ∨ a_{n+1}/a_n ≦ k)] ⇒ Σa_n converges.

https://www.evernote.com/shard/s29/sh/b5fc9d77-98ad-415e-bf5a-c12b66af0577/7442e7245e0720d417c5c0c75ed99748

[56] If z∈C and |z| > 1, then Σz^n diverges.

https://www.evernote.com/shard/s29/sh/46d80a96-1cb9-42ef-95dc-220f1306b0f5/65037777da64feabad96e524519b84f6

[55] If z∈C & |z|＜1, then Σz^n converges to 1/1-z.

https://www.evernote.com/shard/s29/sh/d472bffb-4b6e-42d1-b1d3-fb75f1aee008/cdc8968f534313a7d0cdf695a91db091

[53] Limit comparison theorem／極限に関する比較定理

https://www.evernote.com/shard/s29/sh/70ff849c-7247-493d-8c3c-ea759405624e/c44fe7b65e87788ff92a66ce01faaff5

[52] A nonnegative term series Σa_n converges ⇔ The sequence of partial sums (S_n)n∈N is bounded above.

https://www.evernote.com/shard/s29/sh/4863f174-7629-43d8-be17-672a59ba6ced/c76ada5da43dd99b3bf888581ebf0d15

[51] If Σa_n converges, then a_n → 0 (n → ∞)

https://www.evernote.com/shard/s29/sh/cac04f48-e0ef-4328-ab15-d7e7b21e3a8e/b5a800b9a9bbedea37837f83f08969aa

[50] Cauchy's Convergence Criterion on Series: A sequence of partial sums is convergent iff it is Cauchy.

https://www.evernote.com/shard/s29/sh/778c319e-040b-4bea-b21c-25af5b3c5cf8/00a97e33a2f30933fed6abb42d8ad157

[49] A sequence of points in R^n converges ⇔ It is a Cauchy sequence.

https://www.evernote.com/shard/s29/sh/f09e5c90-c804-4f47-8139-57cdc1d4549f/6c8ee2cae62e18d71a885348b9ba0923

[48] Geometric series／等比数列の和 Σa_k = a(1-r^n) / 1-r (when r≠1)

https://www.evernote.com/shard/s29/sh/f2c30088-c485-424e-8502-fdb56c95c1cf/4d51d96ffbca186c2401ea9966d4b1bf

[47] A sequence of points in R^n (a_m)m∈N converges. ⇒ (a_m)m∈N is bounded.

https://www.evernote.com/shard/s29/sh/20df19ac-ae38-4a8c-9366-1dd515518bb6/17cd6049725c2caf83c05c01da57955e

[45] A sequence of points (a_m)m∈N in R^n is Cauchy ⇔ The sequence of real numbers (a_{m,k})m∈N is Cauchy for 1≦∀k≦n.

https://www.evernote.com/shard/s29/sh/3c65d684-013d-41f8-9aca-98faf68ac62f/5406743f7d4ec6fd7354b80997113ed5

[45] A sequence (a_m) of points in R^n converges to b = (b_1, b_2,..., b_n) ⇔ 1≦∀k≦n, (a_{m,k})m∈N converges to b_k.

https://www.evernote.com/shard/s29/sh/6b698804-7ddf-4cee-909d-64234c384caa/0af43d636e71745c6edb25d992fb26db