A series over the real numbers or complex numbers converges if and only if its tails satisfy: for every , some makes whenever . This says exactly that its partial sums form a Cauchy sequence; sufficiency uses completeness.
If the partial sums tend to a finite limit , then . Thus failure of the terms to tend to zero proves divergence. The converse is false: the harmonic series has terms tending to zero but diverges.
For a fixed positive integer , if and the series of the displayed block sums converges, then the original series converges to the same limit of a sequence. Its partial sum at a block endpoint is exactly a partial sum of ; the difference at an intermediate endpoint has at most terms, all tending to zero. This justification is essential when block cancellation proves convergence without absolute convergence. It does not apply without further control to blocks whose lengths grow.
If the partial sums of are bounded and decreases to zero, then converges. The uniform version holds for functions when their partial sums have one bound independent of . Indeed partial sums starting at index have absolute value at most , and summation by parts bounds every weighted tail from to by . The uniformly Cauchy sequence criterion proves uniform convergence. This justifies integrating a conditionally convergent Fourier series on intervals staying away from its endpoint jump.
For nonnegative sequences with , convergence of implies convergence of , while divergence of implies divergence of .
For positive sequences , suppose eventually. Then is eventually nonincreasing, so beyond a fixed index. The comparison test for series proves convergence of whenever converges. Reversing the ratio inequality gives an eventual lower bound and hence divergence when the positive comparison series diverges. The initial finite set of terms does not affect either conclusion.
If and , clipping every term at preserves divergence. If infinitely many terms are at least , the clipped series has infinitely many terms equal to and diverges. Otherwise its tail agrees with the original series. This lemma is useful in proving convergence from minimum-decrement inequalities.
If is decreasing, then and either both converge or both diverge. This follows by bounding the area under the graph between adjacent upper and lower rectangle sums.
For a nonnegative nonincreasing function , comparison with unit-width rectangles givesOn , the function is at most ; on , it is at least . Summing these inequalities gives the bounds. Applying them to gives for the harmonic numbers. Unlike a convergence verdict alone, the finite bounds estimate the size of a partial sum.
For positive terms, if the ratio of two sequences tends to a finite positive number, their series either both converge or both diverge.
For nonzero terms, ifthen converges absolutely. If the ratio has a limit greater than one, the terms do not tend to zero and the series diverges.
Suppose and, for all sufficiently large , with fixed . Choose an initial index . Since for , summing logarithmic ratios gives . Thus . For , the comparison test for series with a P-series proves convergence. This supplies a useful refinement when a limiting ratio of one makes the ordinary ratio test inconclusive.
For a positive sequence, if , thenFor this follows by taking logarithms and applying Cesàro averaging; the case follows from an eventual geometric upper bound.
