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Theorem 2: Diverging Geometric Series

The Cauchy Convergence Criterion and its corollary can often be used to show that a series diverges. Table 9.2.2 sums up the various possibilities. In this table it is understood that

a1 + a2 + ... + an + ...

is an infinite series and H, K are positive infinite hyperintegers with H < K.

Table 9.2.2 Cauchy Convergence and Divergence Tests

Hypothesis

Conclusion

all aH+1 + ... + aK ≈ 0

Converges

all aK≈0

none

some aH+1 + ... + aK not amost equal to 0

Diverges

some aK not amost equal to 0

Diverges

We shall give many other convergence tests later on in this chapter. For convenience there is a summary of all these tests at the end of Section 9.6.

THEOREM 2

(i) If |c| ≥ 1 the geometric series 1 + c + c2 + ... + cn + ... diverges.

(ii) The harmonic series 09_infinite_series-79.gifdiverges.

PROOF

(i) For infinite H the term cH is not infinitesimal, so the series diverges,

(ii) Intuitively this can be seen by writing

09_infinite_series-80.gif

Instead we can use the Cauchy Test. We see that for each n,

09_infinite_series-81.gif

Therefore for infinite H,

09_infinite_series-82.gif

Since the above sum is not infinitesimal the series diverges. example 3 The harmonic series

09_infinite_series-83.gif

is the example promised in our warning. It has the property that

09_infinite_series-84.gif

and yet the series diverges.
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Last Update: 2006-11-08