Let $(x_n)_{n\in\mathbb N}$ be a real sequence (or a complex sequence). The sequence $(s_n)_{n\in\mathbb N}$ of partial sums $$s_n:=\sum_{k=0}^n x_k,\quad\quad n\in\mathbb N$$ is called the (infinite) series $$\sum_{k=0}^\infty x_k\quad\quad( * ).$$
Note: If the sequence of partial sums is convergent, the expression $(*)$ also denotes the limit (i.e. the real or complex number) to which the sequence converges.
Chapters: 1
Corollaries: 2
Definitions: 3 4 5 6 7 8
Lemmas: 9 10
Parts: 11
Proofs: 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Propositions: 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57
Solutions: 58
