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الكلية كلية التربية للعلوم الصرفة     القسم قسم الفيزياء     المرحلة 2
أستاذ المادة هدى عامر هادي       21/03/2021 20:58:03
In the previous lesson we studied the concept of series. Also we studied series sum and what could it mean by the partial sums of the series. Today we have to take a look on some special series.
Theorem: Let {a_n } and {b_n } be two series of real numbers, the following rules hold
Sum Rule: ?_(n=1)^???(a_n+b_n)?=?_(n=1)^???a_n+?_(n=1)^??b_n ?.
Difference Rule: ?_(n=1)^???(a_n-b_n)?=?_(n=1)^???a_n-?_(n=1)^??b_n ?.
Constant Multiple Rule: ?_(n=1)^??? ca_n ?= c?_(n=1)^??? a_n ?.
If we have {a_n } divergence then ?_(n=1)^??? ca_n ? also divergence.
It is possible that there are two divergent series their sums together gives a convergent series.

Special Series
Geometric Series




2) Telescoping Series
The Telescoping series is a series whose partial sums have a finite number of terms after cancellation. The cancellation technique, with part of each term cancelling with part of the next term, is known as the method of differences.



The partial sums of the series converge to 1. Therefore, the series is convergent and its sum is 1.
Remark: Not every series that can be broken its fractionation is called a telescope series. To get a Telescope series must have to break up a fraction into smaller fractions.
3) Harmonic Series
The harmonic series is the divergent infinite series. Its form is


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