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A geometric series refers to an ordered list of numbers, in which you can find out each term following the first term by carrying out the multiplication of the previous one by the common ratio. An infinite geometric series is the sum of a geometric sequence of an eternal nature. In other words, there is no last term in this series. This series would keep on going on forever. Keep reading these study material notes on infinite geometric series to build a formidable understanding of this topic.

Understanding Behaviour of Geometric Sequences

There is a way to check whether a particular sequence is geometric or not. This involves checking whether successive sequence entries all have the same ratio. A geometric series standard ratio can be negative. If this happens, it results in an alternating sequence. 

An alternating sequence will have numbers whose switching takes place back and forth between signs that are positive and signs that are negative. 

An excellent example of a geometric progression with a -3 common ratio is 1, −3, 9, −27, 81,−243,…

The behavior of a geometric sequence is dependent on the common ratio’s value. One can understand this better with the following properties:

  • If the common ratio is positive, the terms will all have the same sign, just like the initial term.
  • In case the common ratio is negative, an alternation of terms will occur between positive and negative.
  • In case the common ratio is greater than 1, an exponential growth towards positive infinity (+∞) shall occur.
  • You will get a constant sequence progression if the common ratio is 1. 
  • An exponential decay toward 0 shall occur if the common ratio is not 0 but between -1 and 1. 
  • The progression is an alternating sequence that results from a -1 common ratio. 
  • There shall be an exponential growth toward positive and negative infinity for the absolute values if the common ratio is less than -1.

Infinite Geometric Series Formula

The formula is the first thing to study in the study material notes on infinite geometric series.

The infinite geometric series formula is as follows: if the value of r is such that −1 < r < 1, it can be given as,

Sum = a/(1-r)

Here, the r’s value is such that −1 < r < 1. 

Also, in the infinite geometric series:

  • The series first term is a
  • The common ratio is r.

An important point to note is that the common ratio is between two consecutive terms and −1 < r < 1.

Infinite Geometric Series Study Material Detailed Example

The below-detailed example in this infinite geometric series study material will help you get a good grasp on the topic.

Consider a series: 5 + 2.5 + 1.25 + 0.625 + 0.3125…,

The first term in the above series shall be a1 = 5 

The common ratio in the above series shall be r = 0.5.

The series will converge to some value because the common ratio for the above sequence is between -1 and 1.

Now, the summing of the first few terms is as follows:

So, we have a1 = 5

Now, solving further a1 + a1r = 5 + 2.5 = 7.5

Which will in turn become a1 + a1r + a1r2 = 5 + 2.5 + 1.25 = 8.75

Finally, we have = a1 + a1r + a1r2 + a1r3 = 5 + 2.5 + 1.25 + 0.625 = 9.375

Now, the continuation of this pattern can take place to give us the following sums in infinite geometric series:

  • Solving for 5 terms = 9.6875
  • Solving for 6 terms = 9.84375
  • Solving for 7 terms = 9.921875
  • Solving for 8 terms = 9.9609375
  • Solving for 9 terms = 9.98046875
  • Solving for 10 terms = 9.990234375
  • Solving for 11 terms = 9.995117188
  • Solving for 12 terms = 9.997558594
  • Solving for 13 terms = 9.998779297
  • Solving for 14 terms = 9.999389648

Conclusion

An infinite geometric series is the sum of a geometric sequence of an infinite nature. There is no last term in this series, and its continuation will occur forever. The formula of this series is- sum = a/(1-r), where ‘a’ is the first term while ‘r’ is the common ratio. Furthermore, a geometric sequence’s behavior depends on the common ratio’s value. Study the detailed example of this series to make yourself even more familiar with the topic.

Where a × (b × c) ≠ (a × b) ×c

Properties of Vector Triple Product

  1.   A vector triple product yields a vector quantity as a result.
  2.   a⃗ × (b⃗ × c⃗) ≠ (a⃗ × b⃗) ×c⃗ 
  3.   Vector r=a×(b×c) is coplanar to b and c and perpendicular to a.
  4.   Only if the vector outside the bracket is on the leftmost side, does the formula r=a1+λb hold true. If it isn’t, we use the principles of cross-product to shift to the left and then use the same procedure.

Example

  1. Suppose vectors a, b and c are coplanar. Prove that a x b, a x c, and b x c are also coplanar.

ANS: As they are coplanar, we can write them as

[a x b x c] = 0

By squaring both sides, we get:

[a x b x c]2 = 0

[(a⃗ × b⃗) (b⃗ × c⃗) (c⃗ × a⃗)] =0

Therefore, the products are also coplanar.

Difference between Scalar Triple Product and Vector Triple Product

The dot product of a vector with the cross product of two different vectors[3] [SR4]  is called the scalar triple product. For example, if a, b and c are three vectors, the scalar triple product is a. (b x c). The box product and mixed product are other names for it. The volume of a parallelepiped is calculated using the scalar triple product, where the three vectors indicate the parallelepiped’s neighboring sides.

The cross product of vector a with the cross products of vectors b and c is known as their Vector triple product. The vectors b and c are coplanar with the triple product. In addition, the triple product lies perpendicular to a.

Conclusion

The quantity of a vector triple product may be computed by cross-producting a vector with the cross product of the other two vectors. As a result of this cross-product, a vector quantity is generated.

The quantity of a vector triple product may be computed by calculating the cross-product of a vector with the cross products of the other two vectors. As a result, a vector quantity is generated. The BAC – CAB identification name may be acquired from the result after the vector triple product has been simplified.