6. Estimations, series and errors

Using Lagrange theorem to approximate values

Recall that Lagrange Theorem states that for every function differentiable in an interval , there exists in this interval, such that

thus

but for small we can approximate and get the value approximated by the values of and at , so:

Errors and estimating them

If a physical value is calculated using measurements, , where were measured and are equal to with possible deviation of , then we can estimate that the error for is not greater than:

and the maximal relative error equals to .

E.g., if the measurement of the sides of a cuboid was done with observational error of cm and results , , [cm], and then the volume was calculated, then the error can be estimated in the following way. We get , , , which at gives: , and respectively.
So the maximal error is:

so the maximal error is cm, and the maximal relative error (bo ), which is .

Series

Given a sequence of real numbers , we can consider series , simply as a sequence of partial sums:

The sum of a given series is the limit of sequence , if it exists. If it is finite, then we say that the series converges.

E.g. series is convergent and sums up to . Indeed, , which converges to .

Necessary condition

It is easy to notice that if series is convergent then the sequence is convergent to zero. In particular, if is not convergent or converges to a non-zero limit, then cannot be convergent.

E.g. series is not convergent, because sequence is not convergent. Also is not convergent, because converges to .

Notice that the reverse implication is not true. Series may not be convergent even if converges to zero. E.g. sequence .

Arithmetic and comparison criterion

Since series can be seen as sequences, many theorems on arithmetic of limits make sense also in the case of series. For example, limit of sum of two series is the sum of their sums. Notice that multiplying is a bit tricky. Multiplication of elements of series is not the same as multiplication of partial sums.

But we have a criterion which is implied by the three sequences theorem. If for all greater then some number and is convergent then also converges.

E.g. series is obviously convergent, because .

D’Alembert criterion

D’Alembert criterion states what can be said how convergence of depending on limit of . If this limit is , the series is convergent, but if , it is not.

E.g. is convergent, because .

Notice that the criterion does not give any result if the limit equals .

Cauchy criterion

Cauchy criterion states what can be said how convergence of depending on limit of . If this limit is , the series is convergent, but if , it is not.

E.g. is convergent, because .

Notice that the criterion does not give any result if the limit equals .

Leibniz theorem

Leibniz theorem states that if is non-increasing and converges to zero, then is convergent.

In particular, is convergent, since is non-increasing and converges to zero.

Absolute convergence

We shall say that is absolutely convergent, if is convergent. Notice that, if a series is absolutely convergent, then it is convergent.

E.g. is convergent, but is not absolutely convergent. On the other hand, is absolutely convergent (and also obviously convergent).

Taylor’s Theorem

Taylor’s Theorem is a very powerful generalization of Lagrange Theorem which we have studied in the previous term. It gives us a possibility to approximate a function with polynomials.

Let and has all derivatives, up to -th. Let, then:

where is some number between and .

Sum will be called -th order Taylor polynomial of in point , and is the -th remainder term.

Calculating approximations

Taylor’s Theorem enables calculating some approximate values of a function in a given point. E.g., . Its second order Taylor polynomial in is , which for is and it is up to the remainder term, so the possible error is not greater than .

Taylor series

Therefore, if on a given interval , then we can write as a sum of a series. E.g. for is convergent on the whole real line, so the Taylor series of in point (it is easy because ) is:

Such a series but for is called a Maclaurin series.