Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

This is a geometric series.

What is its common ratio?

What is the formula for the sum of a geometric series?

This is a geometric series.

What is its common ratio?

What is the formula for the sum of a geometric series?

It depends on the course, but I would suggest some understanding of elementary analysis would be useful, including some of the major results you are likely to use, e.g. contraction mapping theorem (Banach fixed point theorem), Taylor's theorem, intermediate value theorem, mean value theorem. A basic understanding of solutions to ODEs or a first course in differential equations would probably be essential, as well as standard mathematical methods of calculus. Some basic linear algebra might be useful, too (what will mainly be relevant to you are systems of equations and a few things about matrices and eigenvalues).

Programming experience (e.g. MATLAB, or your favourite mathematical computation package or programming language) would also help.

Hi aternus,

Welcome!

Hi Maya/Natasha,

When we take the square root of something, we ask the question: which positive number can we square to get this number?

For example, we know that the square root of 25 is 5, because 5 is the only positive number which we can square to make 25. We can do the same for any positive number, including 0.

But what about negative numbers? Unfortunately, there is no positive number which we can square to get a negative number. We know this, because the square of any number is always positive (or 0, because 0 squared is 0).

Therefore, we must define a new object, called an imaginary unit, which we will call 'i'. We define i like this:

By doing this, we can now write down what the square root of -9 is. It is 3i, because:

-3i is also a square root of -9, but just as when we were taking the square roots of real numbers, we have a 'principal' square root which gives us only one of the square roots -- in this case, 3i, not -3i.

Your calculator can't deal with imaginary units, which is why it gives you an error message.

Hi Natasha,

Welcome!

iamaditya wrote:

See the graph which Ganesh posted. You will see that it converges down to X-axis and almost touches it at that value.

No, it doesn't ever touch the x-axis. The exponential function is strictly positive: it can't have any roots.

However, if we take , then there is indeed a root. In fact, if we allow complex solutions, there are infinitely many of them, and they are precisely the nth values of the Lambert-W function, . This sequence generates all the complex roots.There is one real root, the so-called Omega constant, . There are several exact forms for the Omega constant, such as the 'power tower':and a nice integral relation is:

Welcome!

Welcome!

Happy New Year!

Monox D. I-Fly wrote:

Wow, are you from Middle East?

I believe anonimnystefy is from Serbia. Orthodox Christians tend to celebrate Christmas on the 7th of January (as per the Julian calendar, which we don't use anymore -- we use the Gregorian calendar). Other countries do the same, like Russia.

Hi bob,

Hannibal Lecter is referring to regular expressions in formal language theory.

"a" denotes the set containing the character a, i.e. a = {a}.

"ε" denotes the set containing the empty string (which has zero length).

"∅" as usual denotes the empty set.

All characters belong to some alphabet A, from which one can derive sentences. For instance, a string of parentheses like '))(((' is a sentence formed from the alphabet {(, )}. Some sentences will be true with respect to one language, but false with respect to others. For example, you might like to write some sentence which claims that every element of a set is either even or odd. That sentence would be true in , but not in , for instance (because we can't establish an order on without introducing a norm, e.g. absolute value). This might seem trivial, but it allows us to distinguish between different structures (in this case, that and are different groups). These two groups are clearly not isomorphic, but we can apply the same principle to prove that two groups might not be. The nice thing about this is that this can work with lots of types of structures, even graphs.The * notation refers to the Kleene star, which can either operate on sets of strings, or sets of characters. If S is some set of strings derived from some alphabet A, then S* is the smallest superset containing ε which is also closed under concatenation. In other words, S* has to contain all possible concatenations of whichever strings it contains, e.g. {ab, c}* = {ε, ab, c, abab, abc, ...}. I'm sure there's some kind of rule dictating which elements are listed first. Unless the set of characters is trivial (i.e. empty set or ε) then the Kleene star applied to S will give us an infinite set (which is always countable). So in this case, b* = {ε, b, bb, bbb, bbbb, ...}.

Another example: ab* = {a, ab, abb, abbb, abbbb, ...}

+ means 'choose either/or'. For example, "a + b" means "choose either a or b". So (a+b)* means "choose either a or b, then take all possible concatenations". In other words, (a+b)* is the set of all possible words you can make from {a,b}.

So aa + b* = {aa, b, aab, baa, bb, aabb, bbaa, bbb, ...}

Hannibal Lecter is asking for the sentence which generates all words containing exactly two instances of the character 'a', which is "b*ab*ab*".

aa + b* does not generate a set containing only characters with exactly two instances of the letter 'a', because it contains the elements b, bb, bbb, bbbb, ... all of which don't have any 'a' in them.

In general, the regular expression b*ab*ab*a...b*ab* (where 'a' appears *n* times) generates all words which contain the letter 'a' exactly *n* times.

Infinite strings are a different story.

Hannibal lecter wrote:

raised to the power of anything won't ever have a root, because the exponential function is always positive. You can see this in the graph that ganesh posted -- it won't ever touch (nor go below) the x-axis.Hi, is there a root for the f(x) = e^-x ???

is it close to 0.571143115080177? or that wrong there is no any root?

please help me

Where did you get 0.571143115080177 from? Are you stating the problem correctly? Or is something else meant by the word 'root' here?

Hannibal lecter wrote:

Hi, how to find the root of the following function by fixed point iteration method ?

-exp(1/(3*exp(t)))/(3*exp(t))

That function doesn't have any roots, so any such iteration will simply diverge. Did you write the function correctly?

Welcome, Tindela!

Hi ganesh,

Could you explain #6278 in a bit more detail?

Thanks,

zetafunc