Variable cv.complexe – Is there a function that can be extended in infinite series of powers with bounded positive coefficients?

Let F be a rational function, is there such an F that can be extended as a series of powers with coefficients of non-periodic positive integers of the following form:

$$ F = sum_0 ^ { infty} a_i x ^ i, a_i in mathcal {N} bigcup 0, exists M space a_i leq M $$.

Any obscure meaning can be clarified by comments and discussions

characteristic co-prime of consecutive number powers

Lemma: For each list of powers of consecutive numbers, prove that at least one of the numbers is equal to their sum.

Please prove this lemma.

pathfinder – How do the ki powers of the qinggong monk's spell interact with the Sensei's Mystic Wisdom function?

the Sensei The archetype of the monk allows, from the 6th level, a monk to use the capacity

Mystic Wisdom (Sur)
On the 6th level, a sensei can use his ability to hint when he spends points from his ki pool to activate a class ability (using the normal actions required for each) so that this ability affects an ally unless 30 feet rather than the sensei itself. …

Do one Qinggong The powers of the ki as a spell – taken to replace class features, such as "Slow autumn" for SLA "True Strike" – count as a "class capacity" for the purposes of "Mystic Wisdom" aptitude?

And if so, have the ability "affects an ally within 30 feet rather than the sensei himself" does SLA spell effects affect the ally, or would they simply get the ability to cast it themselves?

dnd 5th – Is this set of powers balanced?

The frame of my campaign begins with the awakening of the players and their escape after the beginning of the experience. The experiences give them a tattoo that they will discover gradually, assigning them exceptional abilities at a cost.

My general idea is that the power of tattoos increases with the level and allows them to do bigger feats with lower costs while "training". It will also be a driving force of the campaign (they will receive visions and, at first, tattoos will threaten their lives).

I'm trying to create a system that is not too unbalanced (note that the penalty is always applied in full at the beginning of the third round):

  • Levels 1 to 3. They do not control their abilities. I (the DM) control exactly when they use it and how. Usually, provide epic backups on predefined situations to build the story. Penalty: Unconscious for 5 minutes, 15% health points lost.
  • Level 3 to 7. They can summon it once per rest and per 24-hour period. It grants them a triple skill bonus on all rolls of the turn, stackable with any other bonus and a unique skill and advantage on the next turn. Sanction: Stunned for 3 turns, 15% Health was lost.
  • Level 7 to 15. They can invoke it once per rest and per 24-hour period. It grants them a triple skill bonus on all turn shots, stackable with any other bonus, as well as an extra action without a turn during that turn. They acquire an individual skill and benefit on the next turn. Sanction: Stunned for 2 turns, 20% Health was lost.
  • Level 16 to 20: They can summon it once per rest and per 24-hour period. It grants them a triple skill bonus on all turn shots, stackable with any other bonus, as well as an extra action without a turn during that turn. They gain double skill and advantage in the next round. Sanction: Stunned for 1 turn, 30% Health was lost.

The idea is this: they can use it to improve their performance and accomplish incredible feats, but it will cost them dearly and leave them unprotected and weak thereafter, so they have to think about twice before using them and when. Power can be invoked at any time, even in the middle of an opponent's turn, to provide, for example, a bonus for a save throw.

This is my first game on 5th and I use the tattoo as the main story pilot, but I'm afraid that the "powers" do not balance the game. Two questions:

  • Since all players receive this: is it balanced between the different classes / races or will it be said that an assistant will benefit much more than a warrior?
  • Is this balanced in terms of scenario or will it make the players too powerful and therefore they will tend to dominate the fights I will oppose them? What worries me here, is that I can not prepare every meeting assuming that they use it. Ideally, the presence of these powers can turn a difficult encounter into a medium difficulty, but it would be a pity if a "deadly" encounter became easy after using the abilities, so it will probably be out of balance.

Chronicles of Darkness 2nd – When and how do the powers of manifestation of ephemera come into play?

The section on ephemera in the Demon: the descent manual and Chronicles of Darkness manual describes the ephemeral manifestation powers and even gives a map of their interactions. However, I do not see any examples of opponents who actually use the mechanisms of Fettering and Reaching and Anchoring etc. Even the book on Night Horrors does not seem to have any opponents using them. It's like you're reading the rules for chess and the book describes in detail how magic affects the way the pieces interact, but no game of chess you've seen or heard of uses them.

It may be that these powers occupy a central place in other areas such as Geist and Werewolf. If so, can you name a particular manual that gives a good example of their use?

adnd 2nd – A priest from Erik receives ALL the powers of a druid?

In the Birthright Rule Book, on page 74, it says the following about Erik's priesthood:

PW as a druid, no longer move silently, hide in the shadows and empathy of the animals
as a guard three levels higher

Since Birthright, there is more than one church of Erik and the leader of the religion is a regent who can be a npc of level 1 (or a player) and at the 15th level (of PH):

The Great Druid knows six spells of each level (instead of the normal game)
spell progression) and can also cast up to six additional spells
levels, either in one spell or in multiple spells whose levels
total to six (for example, a 6th level spell, six 1st level spelled,
three 2nd level spells, etc.)

Or it is not considered a druid, since it comes from the position, because its servants also receive:

Three of these nine are archdruids who roam the world, acting as his
messengers and agents. Each of them receives four additional spells
levels.

If so, the powers of hierophants are also ignored? Since their origin stems from the status of hierophant and results in a cost of spells:

Beyond the 15th level, a druid never gains new spells (ignores the
Priest spell progression table from this point). Casting level
continues to increase with the experience. Rather than spells, a spell
the powers are acquired.

P.S .: I was not able to create a Birthright tag for reputation reasons (Birthright is a campaign parameter for Ad & d 2e)

calculation and analysis – How to calculate the integral of a given function by a generic series of powers?

I have a function that can not be written through known functions, but it remains a son of God and deserves to be an integral part of it. $ [0, 1] $ calculated.
How can I do this in Mathematica?

I hope there is a way to do it.

Just for illustration, let's say that this function is (where $ n $ is an integer):

$$
int_ {0} ^ {1} mathrm sum _ {k = 1} ^ { infty} frac {(- 1) ^ ku ^ {2k} (2k) ^ {2k}} {(2k)! ^ 2} cot { pi u} sin {2 pi nu} , from
$$

It's part of my complicated formula that I'm working on.

Edit:

The sum of the integrals (instead of integrating the sum) one by one is not an option, because the individual integrals take an indeterminate time to calculate. The given example is close to the real integral, so it should be long to calculate, say, 50 iterations. Plus $ n $Mathematica takes longer to calculate individual integrals.

How to prevent Series from producing powers higher than the required powers by asking for an expansion until a constant term?

I noticed something strange today. Consider the code

series[x + x^2, {x, 0, 2}]

This of course produces the usual serial cut at the second power of x. If I run instead

series[x + x^2, {x, 0, 1}]

I get the series up to x, but if I run

series[x + x^2, {x, 0, 0}]

I receive the series again until the first power of x instead of 0, which I would have guessed. How can I force this behavior?

Beyond Fantasy anima – Prometheum Exxet: Does one of the powers of the object Spellcaster grant Zeon object regeneration?

My Director General and I have a final disagreement on the text of the Stored energy modifier in particular – the text in English is the following.

If the object itself is stored energy (via Zeon Container), the user can use his contained energy to cast the spells that it contains. The transfer rate is 50 Zeon points per round of combat. The article recovers energy as if it was a caster with Magic Accumulation 50. (Prometheum Exxet, 28, it's me who points out.)

Does his reading, that the object has no Zeon regeneration just because it has a magical accumulation, is correct, or does my reading, that "as spell checker "includes ZR, MA, and … such, correct? In addition, are there any other modifiers that seem to give a ZR element on its own? (I ask because I want to create elements that work by themselves, and not having any ZR on any of them very easily prevents it from saving it except to achieve the table & # 39; GM fiat & # 39 ;.)

Dungeons and Dragons – What third party D & D book introduced new rituals giving player characters monstrous parts of the body and their associated powers?

A long time ago, I remember reading a book that introduced new spells that allowed a player to appropriate the abilities of monsters by performing a ritual magic giving him the body parts associated with these powers.

I do not remember the name of the book, but I remember that there was an excerpt from text where someone was trying to gain the lungs of a gorgon to be able to use his petrifying breath weapon . The person died of asphyxiation during the ritual after losing his lungs during the ritual before getting the monster's organ or woke up after a death with a bull's head. made of metal.

In another excerpt, I remembered that an apprentice of a devilish wizard was discovering that his master had been killed by the creation of a human-headed soldier, led by a rat, who was killed. he created experimentally using the magic of the book.

I have vaguely the idea that this book was part of a series detailing variants of the "school third" but I do not remember the topics that would have been covered in the series, so it's possible that I mix it up with a different book.

Does anyone know the name of this book or who published it?