Series - Sequences and Series, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

Series

Now we will investigate what may happen when we add all terms of a sequence together to form what will be called an infinite series. The old Greeks already wondered about this, and actually did not have the tools to quite understand it. This is illustrated by the old tale of Achilles and the Tortoise.

Zeno's Paradox (Achilles and the Tortoise) 
Achilles, a fast runner, was asked to race against a tortoise. Achilles can run 10 yards per second, the tortoise only 5 yards per second. The track is 100 yards long. Achilles, being a fair sportsman, gives the tortoise 10 yard advantage. Who will win?

Both start running, with the tortoise being 10 yards ahead. After one second, Achilles has reached the spot where the tortoise started. The tortoise, in turn, has run 5 yards. Achilles runs again and reaches the spot the tortoise has just been. The tortoise, in turn, has run 2.5 yards. Achilles runs again to the spot where the tortoise has just been. The tortoise, in turn, has run another 1.25 yards ahead.

This continuous for a while, but whenever Achilles manages to reach the spot where the tortoise has just been a second ago, the tortoise has again covered a little bit of distance, and is still ahead of Achilles. Hence, as hard as he tries, Achilles only manages to cut the remaining distance in half each time, implying, of course, that Achilles can actually never reach the tortoise. So, the tortoise wins the race, which does not make Achilles very happy at all. What is wrong with this line of thinking?
Let us look at the di�erence between Achilles and the tortoise:

and so on. In general we have:

Now we want to take the limit as n goes to infinity to find out when the distance between Achilles and the tortoise is zero. But that involves adding infinitely many numbers in the above expression for the time, and we (the Greeks and Zeno) don't know how to do that. However, if we define

then, dividing by 2 and subtracting the two expressions:

or equivalently, solving for s_n:

Now s_n is a simple sequence, for which we know how to take limits. In fact, from the last expression it is clear that lim s_n = 2 as n approaches infinity.

Hence, we have - mathematically correctly - computed that Achilles reaches the tortoise after exactly 2 seconds, and then, of course passes it and wins the race. A much simpler calculation not involving infinitely many numbers gives the same result:

• Achilles runs 10 yards per second, so he covers 20 yards in 2 seconds.
• The tortoise runs 5 yards per second, and has an advantage of 10 yards. So, it also reaches the 20 yard mark after 2 seconds.
• Therefore, both are even after 2 seconds.

Of course, Achilles will finish the race after 10 seconds, while the tortoise needs 18 seconds to finish, and Achilles will clearly win.
The problem with Zeno's paradox is that Zeno was uncomfortable with adding infinitely many numbers. In fact, his basic argument was that if you add infinitely many numbers, then - no matter what those numbers are - you must get infinity. If that was true, it would take Achilles infinitely long to reach the tortoise, and he would loose the race. However, reducing the infinite addition to the limit of a sequence, we have seen that this argument is false.

One reason for looking so carefully at sequences is that it allows us to to quickly obtain the properties of infinite series.

We know (at least theoretically) how to deal with finite sums of real numbers.

More interest in mathematics though tends to lie in the area of infinite series :

What do we mean by this in�nite series,   Define the n^th partial sum, S_n by

This now gives us a sequence, the sequence of partial sums,  The infinite series   is said to converge provided the sequence of partial sums converges to a real number S . In this case we define 

If a series does not converge we say that it diverges. We can then say that a series diverges to +∞ if lim s_n = +∞ or that it diverges to -∞ if lim s_n = -∞. Some texts will indicate that the symbol   has no meaning unless the series converges or diverges  will have no meaning.

Example 6.12   is an infinite series. The sequence
of partial sums looks like:

We saw above that this sequence converges to 2, so

Example 6.13 The harmonic series is

The first few terms in the sequence of partial sums are:

This series diverges to +∞. To prove this we need to estimate the n^th term in the sequence of partial sums. The n^th partial sum for this series is

Now consider the following subsequence extracted from the sequence of partial sums:

In general, by induction we have that that

for all k. Hence, the subsequence  extracted from the sequence of partial sums {S_N} is unbounded. But then the sequence {S_N} cannot converge either, and must, in fact, diverge to infinity.

If the terms an of a_n infinite series   are all nonnegative, then the partial sums {S_n} form a nondecreasing sequence, so by Theorems 6.9 and 6.10  either
converges or diverges to + ∞. In particular,   is meaningful for any sequences {a_n} whatsoever. The series  is said to converge absolutely if  converges.

Example 6.14 A series of the form   for constants a and r is called a geometric series. For r ≠ 1 the partial sums are given by

Taking the limit as N goes to in�nity, gives us that

Example 6.15 [p-Series]For a positive number p

 converges if and only if p > 1:

The exact value of this series for p > 1 is extremely di�cult to determine. A few are known. The first of these below is due to Euler.

If p > 1 then the sum of the series is ζ (p), i.e., the Riemann zeta function evaluated at p. There If p is an even integer then there are formulas like the above, but there are no elegant formulas for p an odd integer.

A series converges conditionally, if it converges, but not absolutely.

Example 6.16 1. Does the series   converge absolutely, conditionally,
or not at all?

2. Does the series   converge absolutely, conditionally, or not at all?

3. Does the series    converge absolutely, conditionally, or not at all (this series is called alternating harmonic series)?

Conditionally convergent sequences are rather di�cult with which to work. Several operations that one would expect to be true do not hold for such series. The perhaps most striking example is the associative law. Since a + b = b + a for any two real numbers a and b, positive or negative, one would expect also that changing the order of summation in a series should have little e�ect on the outcome. Not true.

Theorem 6.15 (Order of Summation)
(i) Let  be an absolutely convergent series. Then any rearrangement of terms in that series results in a new series that is also absolutely convergent to the same limit.

(ii) Let be a conditional ly convergent series. Then, for any real number c there is a rearrangement of the series such that the new resulting series wil l converge to c.

This will be proved later. One sees, however, that conditionally convergent series probably contain a few surprises. Absolutely convergent series, however, behave just as one would expect.

Theorem 6.16 (Algebra on Series)  be two absolutely conver-gent series. Then

(i) The sum of the two series is again absolutely convergent. Its limit is the sum of the limit of the two series.

(ii) The difference of the two series is again absolutely convergent. Its limit is the difference of the limit of the two series.

(iii) The product of the two series is again absolutely convergent. Its limit is the product of the limit of the two series.

The Cauchy product of two series  f real is defined as follows. The Cauchy product is

for n = 0; 1, 2,....