Category: Sequence Theorems

Squeeze (Sandwich) Theorem for Sequences

Consider two sequences $$${x}_{{n}}$$$ and $$${y}_{{n}}$$$. When we write that $$${x}_{{n}}={y}_{{n}}$$$ we mean that corresponding values are equal, i.e. $$${x}_{{1}}={y}_{{1}}$$$, $$${x}_{{2}}={y}_{{2}}$$$ etc.

Algebra of Limit of Sequence

Consider two sequences $$${x}_{{n}}$$$ and $$${y}_{{n}}$$$. When we talk about sum of these sequences, we talk about sequence $$${x}_{{n}}+{y}_{{n}}$$$, whose elements are $$${x}_{{1}}+{y}_{{1}},{x}_{{2}}+{y}_{{2}},{x}_{{3}}+{y}_{{3}}\ldots.$$$. Same can be said about other arithmetic operations. In other words sum of sequences is sequence with elements that are sum of corresponding elements of initial two sequences.

Indeterminate Form for Sequence

When we described arithmetic operations on limits, we made assumption that sequences approach finite limits.

Now, let's consider case when limits are infinite or, in the case of quotient, limit of denominator equals 0.

Stolz Theorem

To find limits of indeterminate expressions $$$\frac{{{x}_{{n}}}}{{{y}_{{n}}}}$$$ of type $$$\frac{{\infty}}{{\infty}}$$$ often can be useful following theorem.

Stolz Theorem. Suppose that sequence $$${y}_{{n}}\to+\infty$$$ and starting from some number with increasing of $$${n}$$$ also increases $$${y}_{{n}}$$$ (in other words if $$${m}>{n}$$$ then $$${y}_{{m}}>{y}_{{n}}$$$). Then $$$\lim\frac{{{x}_{{n}}}}{{{y}_{{n}}}}=\lim\frac{{{x}_{{n}}-{x}_{{{n}-{1}}}}}{{{y}_{{n}}-{y}_{{{n}-{1}}}}}$$$ if limit of the expression on the right side exists (finite or infinite).