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lim n tends to infintiy ( {x} + {2x} + {3x}..... +{nx}/ n2(to the square) )where {X} denotes the fractional part of x?Ans) all no.s are positive or 0. so limit is either positive or 0.........(1)now {x}<=1;{2x}<=1;......{x}+{2x}+....{nx}<=nthat implies lim n tends to infinity{x}+{2x}+....{nx}/n^2 <= lmit n tends to infinity n/n^2;that means lim n tends to infinity{x}+{2x}+....{nx}/n^2 <= lmit n tends to infinity 1/n i.e. 0........(2)from (1) and (2);required limit=0;
Absolute Convergence While we first talked about series convergence we in brief mentioned a stronger type of convergence but did not do anything with it as we didn't have any
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An elliptical galaxy has gravitational boundaries defiend by 9x 2 +16y 2 +144z 2 =144. A black hole at the center of the galaxy is interacting with dark matter producing a radiatio
What is Graphing Statistics explain ? The number of times that an event occurs is called its frequency. One of the ways that you can compare or display different frequencies is
Divides a given line-segment externally in the ratio of 1:2 Construction: i )Draw BX making an actueangle at B. ii) Starting from B, mark 2 equal points on BX as shown in the f
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MAKING CONNECTIONS : you have read about what the ability to think mathematically involves. In this section we shall discuss ways of developing this ability in children. As yo
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Hypergeometric Distribution Consider the previous example of the batch of light bulbs. Suppose the Bernoulli experiment is repeated without replacement. That is, once a bulb is
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