Defn. Let X be a complete normed linear space. Suppose {x_n}_{n = 1}^¥ belongs to X, then the infinite series å_{k = 1}^¥ x_k is said to converge if the sequence of partial sums s_n : = å_{k = 1}ⁿ x_k converges. In this case, å_{k = 1}^¥ x_k : = lim_n®¥ s_n. In particular, this definition applies to the real and complex scalar fields.

Examples: 

1. Let X = IR, then å_{n = 1}^¥ 1/(k² + k) = 1.
   Details:  Set a_k = 1/(k² + k) and notice that a_k = 1/k - 1/(k+1). Observe that the sum telescopes, s_n = å_{k = 1}ⁿ 1/k - 1/(k+1) = 1 - 1/(n+1) ® 1 as n® ¥.
2. Let X be the complex numbers, then å_{n = 1}^¥ a zⁿ converges to s = a/(1-z) if |z| < 1, and diverges otherwise.
   Details:  As we have seen before, |s_n-s| = |a zⁿ/(1-z)| £ C rⁿ for some constant (C = |a|/(1-r)) and r = |z|. We see by the theorem that follows (n-th term test) that the series diverges if 1 £ |z|.
3. Let X = C[0,1/2] and let f_n(x) = xⁿ, then å_{n = 1}^¥ xⁿ converges in X to the function f(x) = x/(1-x).

Proposition. Suppose that å_{k = 1}^¥ x_k and å_{k = 1}^¥ y_k converges in X and c is a scalar, then

1. å_{k = 1}^¥ c x_k = c  å_{k = 1}^¥ x_k
2. å_{k = 1}^¥ (x_k + y_k) = å_{k = 1}^¥ x_k + å_{k = 1}^¥ y_k

Proof. The proof follows immediately from the corresponding properties for sequences applied to the sequences of partial sums.   ^[¯]

Theorem. (Cauchy Test) In a complete normed linear space X, a series å_{k = 1}^¥ x_k converges if and only if for each e > 0, there is a natural number N such that for m > n ³ N || å_{k = n+1}^m x_k ||_X < e.

Proof.  å_{k = 1}^¥ x_k converges if and only if the sequence of partial sums {s_n} is Cauchy. The theorem follows since s_m-s_n = å_{k = n+1}^m x_k if m > n.   ^[¯]

Corollary. (n-th Term Test) Suppose that å_{k = 1}^¥ x_k converges in X, then lim_n®¥ x_n = 0 in X.

Proof. Let n = m-1 in the Cauchy test.   ^[¯]

Corollary. (Weierstrass M-Test) Suppose {f_n} is a sequence of continuous functions on [a,b] which satisfies ||f_n||_¥ £ M_n where å_{n = 1}^¥ M_n < ¥, then the series å_{n = 1}^¥ f_n converges in C([a,b]) to some f; that is, there exists a continuous function f on [a,b] such that å_{k = 1}ⁿ f_k converges uniformly to f as n®¥.

Proof. We apply the Cauchy test to S_n, the n-th partial sum for the series å_{n = 1}^¥ f_n. Note that ||S_m - S_n|| = || å_{k = n+1}^m f_k || £ å_{k = n+1}^m M_k by the triangle inequality. Applying the Cauchy criterion to the right hand side of this inequality, since å_{k = 1}^¥ M_k converges, then completes the proof.   ^[¯]

Note. In the previous theorem, the same proof gives a corresponding theorem for the space of continuous functions (with `sup norm') on W, where W is a compact metric space.

Theorem. (Positive Term Test) Suppose each a_n is nonnegative, then the series å_{n = 1}^¥ a_n converges if and only if the sequence of partial sums is bounded. We then can write å_{n = 1}^¥ a_n = sup_n s_n in the extended sense.

Proof. The sequence of partial sums with nonnegative terms is a monotone nondecreasing sequence which will converge to its least upper bound.   ^[¯]

Defn.  A statement p(n) is said to be eventually true if there exists a natural number N such that p(n) is true for every n ³ N. A statement p(n) is said to be true infinitely often if for each natural number N, there exists n ³ N such that p(n) is true.

Corollary.  Suppose that eventually the terms of the series are nonnegative, then the series å_{n = 1}^¥ a_n converges if and only if the sequence of partial sums is bounded.

Proof.  Eventually the sequence of partial sums are nondecreasing.   ^[¯]