Geometric series

A geometric series is a special mathematical series. A geometric series is the series of a geometric progression. In a geometric sequence, the quotient of two adjacent sequence elements is constant. For valid

A start value of the geometric sequence of 1 and a ratio greater than 1 (here 2 ) gives a divergent geometric series: 1, 1 2, 1 2 4, 1 2 4 8, ..., so together 1, 3, 7, 15, ...

With an identical starting value and a ratio of 1/ 2, however, yields the geometric series: 1, 1 1 / 2, 1 1 / 2 1 / 4, 1 1 / 2 1 / 4 1 / 8, ..., ie 1, 3/2, 7/ 4, 15 /8, ... to the limit.

  • 2.1 Numerical Example
  • 2.2 pension bill
  • 2.3 annuity computation with linear dynamics
  • 2.4 Periodic decimal fractions
  • 4.1 Derivation of the formula for the partial sums
  • 4.2 Derivation of the variants

Calculation of the (finite) partial sums of a geometric series

A series is by definition a sequence of partial sums, the value of the limit value range of the sequence of partial sums. A finite sum is thus a sequential member of the sequence of partial sums. The (finite) sum of the first terms of a series is called so - th partial sum and not as " Partialreihe " etc.

Given is a geometric sequence.

Is the corresponding geometric series.

We can from a new episode

Construct, the -th element is in each case the sum of the first terms of the series, called - th partial sum of. This sequence is said to be the sequence of partial sums. ( The series based on partial sums of a series Strictly speaking, in reverse order defined. The above and the usual notation for the series are the but not here, so we have to reconstruct from it only the sequence of partial sums. ) If it converges, is it defines the value of the series. It applies to the value of series s ( here is not spoken of " limit " ):

In words, the value of the series is defined as the limit value of the partial sums corresponding to their sequence, if it converges, otherwise the number will be referred to as a divergent. If the sequence of partial sums to ( plus / minus ) tends to infinity in this case, you usually write or and says the sequence converges to the improper limit value ( plus / minus) infinity or the series have the improper value ( plus / minus) infinity. ( A formula for calculating the limit follows below.)

By now we denote the ratio of two adjacent links, which is the same for all.

Then for all.

For the - th partial sum, this resulted in:

If, then applies ( derivation see below)

If, as is true

The above applies if the follower members are elements of a unitary ring, thus in particular if there are real numbers.

Related sum formula 1

The partial sum

Has for the result

And for (see Gaussian sum formula )

Related sum formula 2

The partial sum

Has for the result

And for (see power sums)


Numerical example

Given is the geometric sequence

Is associated with and the geometric series

The associated sequence of partial sums is given by


Pension account

Suppose you pay at the beginning of each year, € 2,000 in a bank and the interest rates are at 5% [ie, the interest factor is: 1 (5 /100) = 1.05 ]. How much money you have at the end of the fifth year?

The paid in the first year money will earn interest for five years, you get it at the end of 2000, including compound interest · € 1.055. The paid in the second year money is charged interest only four years, and so on. Overall, then gives a banked amount of

With interest, the capital has thus increased by € 1,603.83. When recalculation of account statements is important to remember that is not rounded mathematically in banking.

For comparison: Would not every year per € 2,000 invested paid but right from the start all the € 10,000 over 5 years at 5 % interest, it would be the final amount

Thus, a capital gain of € 2,762.82.

In general: If the insert at the beginning of each year, the interest factor q and the term of n years, then the final value

Annuity computation with linear dynamics

You pay in contrast to the previous example, not every year a fixed amount, but from the second year of each year € more than in the previous year, then the final value

For example, with € in the first year, every year € more than last year, 5% interest (ie interest factor ) and years duration, then the accumulated amount at the end of the 5th year

In this example, not € 10,000, but 11,000 € were paid, so the profit is € 1,707.65. You pay instead of € in the first year only € one and leaves the other factors being equal (so that one deposits as the penultimate example of a total of 10,000 € ), then the final value is only € 11,547.27, which means you pay the same amount that only at the beginning of fewer, more later, then escape a profit ( opportunity costs).

Periodic decimal fractions

Periodic Dezimalbruchentwicklungen contain a geometric series, which can be converted to the above formulas, again a fraction. For example:

Convergence and value of the geometric series

A geometric series or the consequence of its partial sums converges if and only if the amount of real ( or complex ) number less than unity or her initial term is zero. Converges or the underlying geometric sequence namely to zero:

This is a necessary condition for the convergence of the geometric series. Since for and diverges the basic sequence, in this case is thus also divergence of the series. For the divergence of the geometric series stems directly from the fact that, a term which diverges for and. For the case, the divergence always results as certain divergence ( supra ), in the event more than indefinite divergence. The geometric series converges completely, provided that they converge in the normal manner.

The value of the series in the convergence case arises under that formula above for the -th partial sums by thresholding () for the

Because it is

The last formula is valid even in each Banach algebra, as long as the standard is less than one; in the context of linear operators is also called the Neumann series.


Derivation of the formula for the partial sums

The - th partial sum of the geometric series can be calculated as follows:


Multiplying by gives:

When Equation 2 Equation 1 is subtracted we obtain:

Factoring out:

Parts by supplies for the desired formula for the partial sums:

Derivation of the variants

Using the above formula can be represented by termwise differentiation also finite closed ranks, for

For converge limit formation of the corresponding finite series and the infinite series (hence these are even integrated term by term ):

Similarly for higher powers.